An extended coordinate descent method for distributed anticipatory
network traffic control
Marco Rinaldi (corresponding author)
Research Assistant
KU Leuven
Leuven Mobility Research Centre
Celestijnenlaan 300, B-3001 Heverlee, Belgium
Phone: +32 (0)16 32 27 80
marco.rinaldi@kuleuven.be
Chris M.J. Tampère, Ph.D.
Assistant Professor
KU Leuven
Leuven Mobility Research Centre
Celestijnenlaan 300, B-3001 Heverlee, Belgium
chris.tampere@kuleuven.be
1
Abstract
Anticipatory optimal network control can be defined as the practice of determining the set of control
actions that minimizes a network-wide objective function, so that the consequences of this action are
taken in consideration not only locally, on the propagation of flows, but globally, taking into account
the user’s routing behaviour. Such an objective function is, in general, defined and optimized in a
centralized setting, as knowledge regarding the whole network is needed in order to correctly
compute it. This is a strong theoretical framework but, in practice, reaching a level of centralization
sufficient to achieve said optimality is very challenging. Furthermore, even if centralization was
possible, it would exhibit several shortcomings, with concerns such as computational speed
(centralized optimization of a huge control set with a highly nonlinear objective function), reliability
and communication overhead arising.
The main aim of this work is to develop a decomposed heuristic descent algorithm that, demanding
the different control entities to share the same information set, attains network-wide optimality
through separate control actions.
Keywords: Anticipatory Network Traffic Control, Control Distribution, Distributed Optimization
2
1. Introduction
Road networks have been facing ever-increasing demand, a trend that pushed, in the last 30 years,
researchers worldwide in investing considerable efforts in the attempt of addressing the issues of
network traffic congestion. The characteristics of network supply pose a tight constraint on the
maximum achievable level of service, but it is widely recognized that congestion and/or other traffic
externalities can be reduced through intelligent traffic control strategies, efficiently managing
demand.
Traffic control applications originated as early as in the 1950s, their focus being that of determining
locally optimal settings for single traffic light controlled intersections (Webster, 1958). Over the years,
thanks to the increase in computational power, traffic control policies have evolved more and more
towards coordination, recognizing the impact that separated intersections in the network might have
onto each other, and the importance of taking these effects into account (Wallace et al., 1984). The
late 80s and 90s witnessed a new trend in which, additionally to coordination, also on-line
responsiveness to traffic conditions gained importance, together with research and development
now including traffic control policies tailored to motorways, rather than mainly focusing on urban
intersections (Hunt et al., 1981) (Lowrie, 1990). Finally, in the last decade, coordinated control
policies integrating both urban and motorway have been researched.
In the field of coordinated traffic control, a specific subset is that of anticipatory traffic control
policies. These approaches follow from theoretical works developed in the 70s (Allsop, 1974), and
the key contribution with respect to other control strategies is the fact that the road users’ reactions
to changes in traffic control conditions are taken explicitly into account when determining the
control law. This property yields considerable benefits to the traffic controller, as it can now place
itself in the leading role of a Stackelberg interaction with the road users, allowing for greater
efficiency. A major drawback of all coordinated approaches, anticipatory or not, is the fact that in
order to achieve maximal performance the entire set of controllers on the network should be
coordinated simultaneously.
Model-based coordinated traffic control policies are, indeed, of considerable computational
complexity, due to the possibly very high dimensionality (number of controllers being simultaneously
coordinated) and the ever-growing size of the underlying road network. This complexity becomes
even steeper when considering the anticipatory control subset, where sensitivity information over
the whole road network is necessary to correctly anticipate the users’ behaviour, no matter which
and how many controllers are taken into account.
In order to tackle problems of increasing size and complexity, several decomposition, distribution
and decentralization schemes have been developed for the coordinated traffic control problem in
the last decade. These three words, which might sound very similar, warrant formal definitions,
however in literature no widespread consensus is achieved on their meaning, and thus different
works adopt different notions, e.g. (Katwijk, 2008) (Balaji and Srinivasan, 2010)
3
We therefore specify our chosen definitions for the three words, so to avoid misunderstanding:
-
-
-
Decomposition schemes: frameworks that aim at subdividing a centralized, global problem
into more tractable sub-problems, while still performing centralized computation and
retaining globally valid dynamics of the original problem;
Distribution schemes: frameworks that subdivide a centralized, global problem into simpler
sub-problems, and solve them separately, employing, if necessary, some central mechanism
to ensure that the global dynamics of the original problems are retained;
Decentralization schemes: frameworks that subdivide a centralized, global problem into
simpler sub-problems, and solve them fully separately. Unlike the distributed approaches, no
explicit guarantee that the global dynamics of the original problem are retained, although
they might emerge from the decentralized behaviour;
As we detail in the next section, so far while several researchers have been dealing with developing
simplifications for the coordinated traffic control problem, little effort has been done towards
applying the same ideas to the anticipatory traffic control domain.
In this work, our main objective is indeed to fill this gap, and begin investigating if and how the
anticipatory traffic control problem could be separated into simpler, smaller, eventually distributable
problems. In order to do so, we develop a decomposition scheme, and study under which conditions
this scheme is still able to retain globally valid dynamics and, therefore, yield the same level of
optimality as a fully centralized anticipatory traffic control formulation. In order to ensure
computational feasibility and ease of understanding, we perform this approach within the static time
domain, although our focus for future research is that of extending these findings to the field of
within-day dynamics.
The of this paper is organized as follows: in Section 2 we present an extensive literature review; in
Section 3 we detail our methodology; Section 4 will feature the implementation we developed to test
our algorithm, together with a simple proof of concept scenario to show its convergence properties.
In Section 5 we present numerical tests of this algorithm on different size networks, ranging from
simple examples towards bigger instances aiming to real-life, discussing the scalability of our
approach and comparing it with general purpose optimization algorithms. Finally, in Section 6, we
gather conclusions and give insights towards future research topics.
2. Literature Review
For the purposes of this work, we split the traffic control literature in two main categories, and focus
mainly on reviewing the second: local traffic control strategies and coordinated traffic control
strategies.
Local strategies, such as the Webster formula (Webster, 1958) for urban intersections or, considering
motorway control, the ALINEA ramp metering strategy (Papageorgiou et al., 1991) are based on
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locally measured data (either for online or offline purposes) and disregard any interaction with other
controllers deployed on the network.
Coordinated traffic control strategies, instead, take multiple controllers into account simultaneously,
and try to obtain an optimal set of solutions for the resulting combined problem. Several coordinated
strategies have been developed over the years, starting from fixed time area based strategies such as
the MAXBAND (Little et al., 1981) and TRANSYT-7F (Wallace et al., 1984) control policies, and then
evolving so to include responsiveness to online traffic states, such as in the SCATS (Lowrie, 1990),
SCOOT (Hunt et al., 1981) and UTOPIA/SPOT (Mauro and Taranto, 1990) control policies.
Advances in the field of traffic modeling enabled in the 80s and 90s the development of simple
model based coordinated traffic control policies, such as the OPAC (Gartner, 1983), PRODYN (Henry
et al., 1984) and RHODES (Sen and Head, 1997) systems. More recently, thanks to the latest
improvements and developments yielding more detailed, faster and computationally more efficient
traffic flow models, coordinated control strategies based on the Model Predictive Control (MPC)
framework were researched in several works, such as (Hegyi et al., 2005), (Dotoli et al., 2006), (Van
Den Berg et al., 2007) and (Aboudolas et al., 2009).
Anticipatory traffic control policies are a subset of the coordinated control policies, which not only
employ a model to predict the future conditions of the network but also explicitly consider the user’s
route choice reactions. These policies have received considerable attention from researchers
especially in terms of analytical optimality and properties. Approaches as early as that of Allsop
(1974) have been exploring the conditions of existence and uniqueness of solutions to this problem
in the static domain. The impact of anticipatory control policies in static scenarios has been explored
by several authors, dealing with both road pricing (Yang and Bell, 1997) (Yan and Lam, 1996) and
signalized intersection control (Smith and Ghali, 1990), (Cantarella et al., 1991), (Yang and Yagar,
1995).
In recent years, this concept has then been extended to the Dynamic Traffic Assignment (DTA) field
by Taale, (2008) , Taale and Hoogendoorn (2012) and Taale and Hoogendoorn (2013). In their papers,
the authors outlined this approach in a generic computational framework, through a bi-level
formulation, discussing the implications of anticipatory control theory and its computational
feasibility when expanding the problem to the time variant domain. More recently, (Ukkusuri et al.,
2013) proposed an algorithmic model to solve the combined dynamic equilibrium based traffic signal
control problem, which is expressed both as Nash-Cournot and Nash-Stackelberg games, based on an
Iterative Assignment Optimization (IAO) procedure, and assessed the solutions’ quality in terms of
optimality.
As we discuss in the introduction, all coordinated control strategies, anticipatory or not, share a
common difficulty to be tackled, being the considerable complexity arising from the need to align
several controllers simultaneously into one optimization problem. Moreover, in practice traffic
controllers are separated, geographically and/or hierarchically, with local influence to only subsets of
the whole network. Indeed, the effort necessary to develop an architecture whose centralization
level would be sufficient to attain full global optimality is often impractical, with robustness,
communication and computational complexity being only some of the problems such a centralized
system would face.
5
In this respect, several decomposition and distribution schemes have been developed for the
coordinated traffic control problem; some works have focused on subdividing the problem by
reducing the size of the underlying network, subdividing it into more tractable subnetwork, such as in
the approaches of (Papageorgiou and Mayr, 1982), (Gartner et al., 2001), (Mirchandani and Head,
2001), (Kotsialos and Papageorgiou, 2004), (Wongpiromsarn et al., 2012) and (Boillot et al., 2006).
Other authors have been focusing on reformulating the coordinated control problem in
mathematical structures enabling the authors to exploit very fast and efficient optimization
algorithms to determine the problem’s solution. Such works include, for example, the TUC control
policy (Diakaki et al., 2002), the works of (Aboudolas et al., 2009), (Aboudolas et al., 2010), who
employ a store-and-forward traffic flow model formulation to compute optimal green split and
offsets in a network-wide scenario, exploiting the Rolling Horizon principle for online purposes and,
finally, the approaches of (Lin et al., 2012), (Lin et al., 2011), and (Hajiahmadi et al., 2015) where a
Model Predictive Control based problem is also redefined as a Mixed Integer Linear Programming
problem, allowing for fast optimization.
Other authors have been decomposing the problem by exploiting computer scientific programming
frameworks, such as agent-based modeling. Such papers include the store-and-forward model based
approach of (de Oliveira and Camponogara, 2010) and the works of (Wang, 2005), (Katwijk, 2008),
(Balaji and Srinivasan, 2010) and (Du et al., 2014).
In all of these coordinated traffic control decomposition strategies, though, no explicit modeling is
included to take the users’ routing response into account, and therefore, while a few strategies can
achieve, e.g. based on some area-aggregate measure, a redistribution of traffic and queues over the
available capacity of the network, such as in the approaches of (Geroliminis et al., 2013) and (Haddad
et al., 2013), no full-fledged anticipation is considered. A few studies dealt with decomposing User
Equilibrium formulations, such as in the works of (Suwansirikul et al., 1987) and, more recently, the
algorithms developed by (Dial, 2006), (Gentile et al., 2004) and (Bar-Gera, 2010); other focused,
instead, on developing centralized algorithms to solve the road pricing problem efficiently (Dial,
1999), (Bar-Gera et al., 2013).
As a first step towards developing decomposition and, ultimately, distribution schemes for the
anticipatory traffic control problem, in (Rinaldi et al., 2013) we employed a Bi-Level formulation and
an algorithmic solution, also based on the Model Predictive Control framework, and introduced a
controller-wise decomposition scheme; while recognizing the importance of decomposing the
problem within the time domain, through the Rolling Horizon technique, we wanted to explicitly
consider the fact that controllers are, in real-life, pertaining to separate, distributed entities.
Following the empirical results we obtained in that work, in this paper we explore the convergence
properties of the controller-wise decomposition strategy in-depth, through simpler static scenarios,
from a theoretical, analytic and algorithmic point of view; we once again express the anticipatory
control problem as a Bi-level optimization formulation, in which the upper level problem is that of
selecting optimal control values for either traffic light controlled intersections or pricing on a subset
of the network’s links, while the lower level problem is Static, Deterministic User Equilibrium.
The main aim of this work is establishing the conditions under which anticipatory network-wide
control would maintain global optimality while decomposed. Our long term objective is to obtain a
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solid methodology for real-life deployable anticipatory traffic control, determining shortcomings (if
any) in terms of optimality with respect to a fully centralized approach.
3. Methodology
This section outlines the methodology with which we tackle the problem of decomposing
anticipatory optimal control. It is divided into subsections: we first deal with how the information
gathering process can be performed by separated entities, thanks to a sensitivity-wise objective
function reformulation; we then detail the decomposition of the optimization problem into different
controlling units and how we obtain the necessary sensitivity approximations. Finally, we develop an
optimization algorithm tailored to our problem and discuss its convergence properties and
implications.
3.1 Objective function reformulation
When dealing with anticipatory control, in order to determine the optimal control law we exploit the
availability of some measures (be it traffic lights, pricing, ramp metering, variable speed limits, … ) to
meet a network-wide goal, usually expressed in the form of an objective function. “Optimality” is to
be understood in a broad sense; it could for example be related to emission levels, environmental
constraints (Lin et al., 2014), more desirable distributions of traffic over the network’s hierarchy, etc.
In this paper we focus on maximizing the usage of the network’s capacity. This objective can be
formulated, in a static setting, as the minimization of the Total Cost function.
For a generic network N , composed of a set of n arcs L  [l1 ,..., ln ] and a set of nodes V , this
function is defined as follows:
TC N ( g )   f l  g, cl ( g )   cl  f l  g  , g  l  L
(1)
l
where fl (·) and cl (·) are, respectively, the flow and cost functions for each link l , and
g  [ g1 ,..., g m ] is the control inputs vector. Flow and cost functions are recursively dependent due
to the users’ equilibrium response. Following Wardrop’s 1st principle, this interdependency can be
modeled through the following Variational Inequality (VI):
 c ( f )· f   c ( f )·x x  X
l
l
l
l
l
l
l
l
f
(2)
where X f is the set of all feasible link flows.
Under interior point existence assumptions (Verhoef, 2002), the anticipatory traffic control problem
(1) can be formulated as the following optimization problem:
min TCN ( g)
g
7
(3)
and its solution can be obtained by solving the following equation:
g : TCN  g   0
(4)
In order to solve the problem (3), one would need to determine the objective function’s gradient
TCN  g  for all controllers. The necessary sensitivity information would need to be collected from
a fully centralized perspective though, while in this work our objective is that of assessing if and how
the anticipatory control problem can be decomposed in a controller-wise fashion.
We therefore devise a scheme that allows us to separate the sensitivity retrieval process so that part
of it can indeed be performed controller-wise, while a simple and independent monitoring system
can be employed to estimate the remaining portion.
Following the sensitivity analysis algorithm proposed in (Yan and Lam, 1996), we apply a simple chain
rule derivative formula to our problem, thus rewriting the 1st order derivative of the original
objective function (1), explicitly separating three different sensitivity terms:
TCN  g   J Tf  g  TTC  f   [ f ·c( g)]T
(5)
where f  [ fl1 ,..., fln ] is the link flow vector, the Jacobian matrix J f  g  is the sensitivity of
equilibrium flows to the control actions and the gradient TC  f  is the sensitivity of the objective
function (in this case Total Cost, but this can be generalized to any cost function) to changes in the
network flows. Finally, c ( g ) represents the sensitivity of link costs to control, having nonzero
values only when a controller gi has direct influence on the link cost function, as, for example, in
case of traffic light control.
These three objects are defined as such:
 fl1 / g1

J f  g 
 f / g
1
 ln
fl1 / g m 


fln / g m 

TC  f   TCN / fl1
 cl1 / g1

c( g )  
 c / g
1
 ln
8
TCN / fln
cl1 / g m 


... cln / g m 
(6)

(7)
...
(8)
In simple, linear cases these quantities can be determined analytically; minimizing the resulting Total
Cost function would then, under existence and uniqueness assumptions, yield the optimal value for
the control action. It’s worth mentioning that the reformulation of eqn. (5) is not limited to the Total
Cost objective function, but can be applied to other, not necessarily linear or even link-additive
objective functions.
This objective function reformulation follows the idea that different, separate entities could be
responsible for estimating the separate sensitivities, overcoming the need for one centralized
monitoring and control system.
J f  g  and c( g ) are quantities that we envision being estimated by the controllers themselves,
through sensitivity analysis, following e.g. on-line approaches such as that of Zhang and Roberts
(1990), which could be applied on a day-to-day basis. These terms contain, respectively, the whole
network flows’ sensitivity to a change in control, and the local cost sensitivity to a change in control.
From a centralized optimization perspective, the full Jacobian matrix J f  g  and cost gradient
c( g ) are necessary to determine a descent direction for problem (3). If controllers are being
optimized entirely independently of each other, though, as is in the application we discuss in the
later stages of this paper, this sensitivity information can be obtained and employed entirely
separately (controller-wise), as one controller needn’t know the sensitivity of another’s. This means
that each separate controller gi only needs to estimate and know its own column J f ( g i ) .
TC  f  is, instead, a quantity proper to monitoring systems; this sensitivity can be learned, over
time, simply through data observations and statistical analysis; geographically separate portions of
the network might be monitored by different entities, but as long as the variable of interest (flows) is
being measured and stored, together with the respective variations in the chosen objective function,
this sensitivity remains easily obtainable. This latter property is especially true for the Total Cost
objective function, whose dependency on flows is entirely separable (sum over all links). When
dealing with non-separable functions, the estimation process will increase in complexity, as the
separate monitoring entities might need communication in order to assess the impact on common
portions of the objective function. Independently from the chosen optimization framework,
decomposed or centralized, all controllers need full access to TC  f  , as knowledge about the
influence over the whole solution space is needed in order to pursue network-wide goals.
We assume in this work that changes in Total Cost are solely dependent on link flows, we recognize
that there might very well be a case in which this function is influenced by other traffic phenomena,
such as changes in speeds or densities in the network. Our sensitivity decomposition approach could
still be applied; its formulation should simply be extended by adding relevant terms to eqn. (5) so to
account for this extra source of influence.
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3.2 Separate sensitivities approximation
As discussed in Section 3.1, we reformulate the objective function in terms of separate sensitivities in
order to clearly address the problem of obtaining relevant data to compute them. We here introduce
such a mechanism, in which we obtain linear approximations for the two sensitivity matrices by
truncating their Taylor Series at the first polynomial degree. Through this approach the response of
the traffic system to changes in control can be determined even when the two sensitivities are not
analytically known or easily computable, as is for example when dealing with nonlinear link cost
functions; linear approximations to these sensitivities can then be obtained through finite differences.
Finite difference estimation of network sensitivity to control has been developed for real-life
applications, e.g. by applying predetermined perturbations to the control variables (Roberts, 2000) or
by using measurements observed in previous time steps (Brdys and Tajewski, 1994) to reconstruct
the local approximation.
Throughout this section the approximations of variables and derivatives are marked with a (·)
symbol.
Based on the objective function reformulation introduced in Section 3.1, we can write a first order
approximation of the Total Cost function as follows:
TC N  g   TCN  g0   TC N  g    g  g0 
(9)
with
TC N  g   Jˆ Tf  g  TC
T
 f   [ f ·c( g)]T
(10)
and g 0 the initial control vector value. In the case of our link-additive objective function (1), we can
also express this approximation as a sum of its link-wise components:
TC N  g   TCN  g0   (fl  g  TC  fl    g  g0   fl ·cl ( g )·( g  g0 ))
(11)
lL
We choose to approximate the three separate sensitivity matrices J f  g  , TC  f  and c ( g )
linearly, through their Taylor Series, as follows:
the columns composing J
f
 g  are in the form f  g  , which is a value immediately obtained
from the 1st order approximation of the flow function:
f  g   f ( g ) g   g  g0   f 0
(12)
0
Where f 0 is the initial flow vector and the gradient f ( g ) g is evaluated based on the full
0
network’s equilibrium response, around the initial control vector value.
10
The elements composing TC ( f ) , instead, need some elaboration:
For a fixed value of the control vector g , the Total Cost for a network N has the following form:
TCN  ( fl (cl )  cl ( fl ))
(13)
lL
any given element of TC ( f ) would bear the following form:
TCN
c( fl )


fl  cl  fl    cl  fl  
 fl

fl
fl
fl
(14)
in order to obtain an approximate value for these elements, the cost functions of the links can first
be linearly approximated with respect to changes in their flow f l as follows:
cˆl  fl  
cl
f l


 fl  fl0  cl0
(15)
f0
where f l0 is the initial flow for link l , and cl0 is the resulting link cost. Substituting this
approximation into equation (14), we obtain the following formulation for the linear approximation
of the link-wise total cost’s response to changes in flow:
TC N cl

fl
fl


 fl  fl0 
f0
cl
f l
 fl  cl0
(16)
f0
Finally, the linear approximation of the gradient of link costs with respect to changes in control is
very straightforward, as its elements can be determined by taking the 1st order approximation of
costs:
cˆ( g )  c( g ) g   g  g0   cl0
(17)
c( g )  c( g) g
(18)
0
it follows naturally that
0
Summarizing, the elements of the linear approximations for the sensitivity matrices, presented here
in a compact vector form, are obtained as follows:
 f l ( g ) 
g0 
 1


J f  g 


 fln ( g ) 
g0 

11
(19)
TC  c( f ) f   f  f 0   c  f  f  f  c0
(20)
c( g )  c( g) g
(21)
0
0
0
As we will introduce in the next section, this choice of approximation neatly connects with how we
plan to tackle the control-wise decomposition of the problem, yielding a quadratic approximation for
the objective function.
3.3 Controller-wise optimization procedure decomposition
In (Rinaldi et al. 2013) we explored the possibility of decomposing optimal, network-wide control by
minimizing a network-wide objective function J (·) separately for each controller, given global
information. In an effort to characterize the properties of this method, we analyze its convergence in
detail when dealing with a static objective function, which we discussed in Sections 3.1 and 3.2.
The practice of minimizing an objective function separately, variable by variable, is well-known in
optimization, within the framework of Derivative Free Optimization (DFO) algorithms (Conn et al.,
1997), and is basis of “coordinate descent” methods.
The most well-known variant of coordinate descent methods, usually referred to as “coordinate
search” method, performs a search along a specified direction, starting from the current iterate, for a
point that provides a “sufficient” descent for the objective function J ( x1 ,..., xn ) where
x  [ x1 ,..., xn ] is the vector of decision variables. This method cycles through the 𝑛 coordinate
directions [e1 ,..., en ] of x and obtains new iterates by performing a line search procedure, along
each direction, in turn.
At the first iteration, all components of x , save for x1 are fixed, a new value that reduces
(minimizes) the objective function along that direction is sought. On the next iteration, we repeat the
process for the second component x2 and so on. DFO algorithms operate under the assumption that
the analytical expression of the objective function’s gradient is unknown, and therefore either
approximate it numerically, or, as is usually with coordinate descent, seek for a descent direction
through direct search methods, such as golden section search, ignoring any derivative information
(Fletcher, 2013). In general, the convergence of this method is less than satisfactory, highly
dependent on the chosen line search procedure, although it can be guaranteed under some
assumptions (with convexity being a major constraint) as discussed in (Luo and Tseng, 1992).
Recently, convergence for objective functions that are a combination of convex and non-smooth
separable functions has also been discussed in (Tseng and Yun, 2009).
In order to overcome or, at least, reduce the impact that these convergence limitations would pose
on the method’s applicability, we study another class of methods, referred to as “Model-Based
Methods”. These consist of constructing a linear or quadratic model of the objective function around
the current working point, and obtaining the next iterate by minimizing this model within a trust
region. This model is usually defined as the quadratic function mk (·) that interpolates the original
12
J (·) in a set of well-chosen sample points, and the first and second order derivatives are defined
based on interpolation conditions. Once the model has been formed, these methods employ a trustregion sub-problem to approximate the descent step p , and move to the next iterate if a sufficient
reduction in the objective function has been met (Wright, 1999).
In order to improve the convergence rate of Coordinate Descent, we combine its properties with
Model-Based methods: at the k -th iteration, rather than performing a line search over the objective
function, possibly disregarding any derivative information, we build a quadratic model of the
objective function J ( x1 ,..., xn ) for the i -th coordinate direction ei around the current value xk ,i ,
and choose the step pi that minimizes this model in a trust-region with radius  k , obtaining then
the next iterate: xk 1,i  xx ,i  pi . All other components of x are left unmodified. Other than
improving the convergence rate by exploiting some approximation of derivative information, this
method also takes in to account the fact that, in this application scenario, objective function
evaluations tend to be quite expensive (a full Static Assignment procedure is needed each time), and
that therefore a line search procedure based on function evaluations alone would be
computationally cumbersome.
Given the approximation for the first order derivative of the Total Cost function introduced in Section
3.2, it’s immediate to realize that obtaining a quadratic model for the i -th coordinate direction ei
around the current control point g 0,i is straightforward:
TC N  gi   TCN  g0,i   (fl  gi  TC  fl    gi  g0,i   [ fl ·cl ( gi )]T ·( gi  g0,i ))
(22)
lL
This means that the only needed objective function evaluations are those necessary to compute or
update the linear approximations for the three sensitivity matrices J f ( g i ) , TC ( f ) and cl ( gi ) .
As we will detail in the next section, our method of choice is a central finite differences scheme,
meaning that two function evaluations per controller are needed.
The size of the trust-region will be updated based on a metric assessing the goodness of the
quadratic model mk : the more regular the shape of the objective function around the current point,
the larger steps our approach will be able to take (Wright, 1999).
Finally it’s worth noticing that, if the objective function J (·) is convex, minimizing the function along
the coordinate directions ei introduces no error with respect to optimizing it against the full variable
set; the only error introduced by this scheme is then that of the approximated model of choice.
13
3.4 Full Algorithm
In this Section, we first develop and outline a decomposed algorithm for solving the bi-level pricing
subject to user equilibrium problem, combining the different methodological steps introduced earlier
in this chapter, and then discuss its inner workings in full detail together with clear definitions of its
parameters, its convergence conditions and, finally, the implications of our choices with respect to
developing large-scale control distribution mechanisms.
Definitions of parameters employed in the following algorithm:
g0
Initial condition vector
k

 Max
k
Trust region radius at iteration k
Trust region update threshold
Maximum trust region radius
Iteration counter
The trust region radius is controller-dependent, and has the same units as those of the specific
controller.
Algorithm 3.1:
Choose an initial point 𝑔0
Choose an initial value for the trust region radius Δ0 , an update threshold 𝜂 and a
maximum trust region radius Δ𝑀𝑎𝑥
1. Set 𝑘 ← 0
2. repeat until a convergence test is satisfied:
3. compute 𝒇𝒌,𝟎 (Equilibrium Assignment)
4. for each coordinate direction 𝑒𝑖 in 𝒈:
̂
i. compute 𝐽̂𝑓 (𝑔𝑘,𝑖 ), ∇̂
𝑇𝐶 (𝒇) and ∇𝑐 (𝑔𝑘,𝑖 ) around 𝑔𝑘−1,𝑖 and 𝒇𝒌,𝒊−𝟏
̂
ii. form the quadratic model 𝑚𝑘 (𝑔𝑘,𝑖 ) = 𝑇𝐶𝒩,𝑖 (𝑔𝑘,𝑖 ) + (𝑔𝑘,𝑖 − 𝑔𝑘−1,𝑖 ) ⋅ ∇𝑇𝐶
𝒩,𝑖 (𝑔𝑘,𝑖 )
𝑇
̂
̂
̂
̂
(𝒇)
where ∇𝑇𝐶𝒩,𝑖 (𝑔𝑘,𝑖 ) = 𝐽𝑓 (𝑔𝑘,𝑖 ) ⋅ ∇ 𝑇𝐶
+ 𝒇 ⋅ [∇𝑐 (𝑔𝑘,𝑖 )]
iii. solve min 𝑚𝑘 (𝑔𝑘,𝑖 + 𝑝𝑖 ) 𝑠. 𝑡. ‖𝑝𝑖 ‖ ≤ Δ𝑘
𝑝𝑖
iv. update Δ𝑘
v. if 𝑝𝑖 meets satisfactory criteria
Set 𝑔𝑘,𝑖 ← 𝑔𝑘−1,𝑖 + 𝑝𝑖
else
Set 𝑔𝑘,𝑖 ← 𝑔𝑘−1,𝑖
vi. compute 𝒇𝒌,𝒊 (Equilibrium Assignment)
5. end(for)
6. Set 𝑘 ← 𝑘 + 1
7. end(repeat)
After the initialization, the algorithm performs an iterative search of successive steps updating the
full control vector gk , but each component g k ,i is updated separately over an inner loop iterating
through the coordinate directions ei . Each outer iteration begins by computing a full equilibrium
14
assignment around the current value for the full control vector g k 1 , which will serve as a basis for
the inner iterations. The inner iteration loop performs the following operations:
i.
the linear approximations for the sensitivity matrices are computed around the previous
iteration’s i-th control and the current flow pattern; all three are obtained through finite
central differences, with J f being computationally quite expensive (two full equilibrium
assignment procedures), while the latter require only small link-wise perturbations in order
to compute the link cost function’s sensitivity around the current flow and control patterns.
The columns of J f not pertaining to controller i are simply left unmodified.
ii.
based on the updated sensitivities, the quadratic model mk is computed (see Appendix A for
iii.
a detailed formulation).
we minimize the quadratic model within the current trust region, and obtain a candidate
step pi
iv.
the trust region radius  k is updated.
v.
the candidate step is either accepted or refused.
vi.
finally, the network is re-equilibrated based on the new value for g k ,i , the rest of the control
vector is kept at the previous iteration’s values.
Both steps (iv.) and (v.) are based on computing a metric that measures the goodness of the
quadratic model mk , and how well it represents the real function’s behaviour. We choose a metric
defined as follows:

TCN ,i  g k ,i   TC N ,i  g k ,i  pi 
mk  g k ,i   mk  g k ,i  pi 
(23)
Given a threshold  , the trust region size update scheme is as follows:
Δk  min  2  Δk ,ΔMax  , ρ  η
(24)
1
Δ k   Δ k , ρ  η
4
(25)
The candidate step 𝑝𝑖 , instead, will be accepted if   
This is a common approach in trust region algorithms:  is an indicator of the ratio between the
actual reduction in the objective function and the reduction predicted by the model. The closer this
indicator to one, the better the model is a fit for the real function, and therefore the more “trust”
can be put into it. In a similar fashion, if the model can be trusted, the resulting descent step can also
be accepted. While the  k parameter gets updated over the iterations by the algorithm, the 
parameter is an exogenous input – depending on the regularity and convexity of the function, one
might choose different values for it, with risk-averse values being those close to 1.
15
In order to complete this discussion, we now introduce two propositions regarding the convergence
properties of our algorithm, following from simple assumptions on the objective function’s
properties.
Proposition 3.1:
Given the objective function TCN  g  , if, along its coordinates ei  [e1 ,..., en ] , the Total Cost
components TCN ,i  gi  are quadratic in gi i , then Algorithm 3.1 will terminate in a finite number
of iterations and the sequence of iterates {gk ,i }  gi i .
*
Proposition 3.2
Given a convex objective function TCN  g  , under the assumptions that, at the k-th iteration:

the link cost functions cl ( fl ) are twice differentiable on the open interval, with cl ( fl )
continuous on the closed interval between f l , k and fl ,k 1

the i-th separate Total Cost function component TCN ,i  gi  is twice differentiable on the
open interval with TC N ,i ( g i ) continuous on the close interval between g k ,i and g k 1,i
the approximation error of the quadratic model mk ( g k ,i ) of the separate Total Cost component
TCN ,i  gi  is, following Taylor’s theorem’s application to our approximation, bound by
 g
k ,i
lL
 g k 1,i   R1c  fl ,k   R1TC  g k ,i 
(26)
where
R1c  f l ,k  
 2 cl  L 
2!
f
l ,k
 f l ,k 1 
2
(27)
is the second order term of the Taylor series expansion of the link cost function in mean-value form
with  L  [ f l ,k , fl ,k 1 ] and
TC
1
R
g  
k ,i
 2TC N ,i  L' 
2!
g
k ,i
 g k 1,i 
2
(28)
the second order term of the Taylor series expansion of the Total Cost function in mean-value form
with  L '  [ g k ,i , g k 1,i ]
when considering errors higher than second order to be negligible.
16
The trust region update scheme discussed earlier guarantees that the step size remains small, unless
the quadratic model is considered a good fit for the original function.
Following these propositions, we derive the conditions under which this scheme yields global
optimality:
1.
2.
3.
4.
5.
6.
Link cost functions are separable
Link cost functions are monotonic
An interior point solution for the optimization problem exists
The chosen objective function is unimodal (i.e. uniqueness & existence of a minimum)
All local entities collaborate to the gathering of cost sensitivities
Network-wide equilibrium sensitivity to control is known, either exact or approximated
3.5 Final remarks
In an effort to clarify the rationale behind the objective function reformulation and the decomposed
optimization algorithm, we discuss an example showcasing how we envision this combined
distributed monitoring and control system would operate in a real life scenario.
In practice, various entities take care, for example, of managing the intersection timing for different
portions of an urban area, while a separate authority, possibly residing in a completely different
geographical location, would instead be dealing with a Ramp metering (RM) system onto an adjacent
motorway (Carlson et al., 2014). The main concern, and clear motivation behind our line of research,
is that when one system (usually, the motorway) is dealing with high levels of congestion, the
consequences might resonate also through the adjacent urban network. Given such a scenario, we
believe that coordinating the urban intersections together with the RM system on the motorway
could help mitigate congestion effects on both systems.
Implementing our approach in this scenario would require two separate stages: first we need to
define the different monitoring entities and the way they communicate. Let’s imagine that, in this
instance, the urban administration takes care of monitoring its own traffic state, but that this system
is indeed independent from the traffic light servers. As for the motorway, we pose that the entity
responsible for the RM system obtains its data from a third party private traffic data provider.
Once the monitoring system is defined, we need to set up the coordination mechanism between the
different controllers; this could be achieved by an additional entity, whose geographical location is
irrelevant, that takes care of:

communicating to the relevant monitoring systems, in order to acquire measurements
necessary to estimate the sensitivities J f , TC and c

computing and relaying this sensitivity information to both the traffic light and the RM
servers
The different players would still operate entirely separately, each optimizing its own part of the
objective function (or, rather, the relevant quadratic model); thanks to the shared sensitivity
information, though, the overall performance would point towards global optimum.
17
The aforementioned example helps to sketch how our approach could fit in one specific real-life
(partial) control distribution instance, but it can be just as easily applied to any other control
distribution nature, e.g.:




distribution according to type of control: a traffic center having separate servers for traffic
signals, for motorway access control (ramp metering), route guidance etc. but wishing to
coordinate actions between these servers
distribution according to hierarchy or geographical authority: a national motorway road
authority interacting with various regional and urban road authorities, wishing to co-ordinate
their traffic control actions
alignment of various levels of control: pricing and operational traffic control with a common
objective function
…
In this work we choose to explore the optimality conditions and feasibility of a full decomposition
scheme, in which all controllers might be optimized separately while sharing common information.
Naturally, the same scheme might be applied to instances in which only partial decomposition occurs,
as would be for the scenario we just discussed. In fact, in (Rinaldi and Tampère, 2015) we developed
a time dynamic version of this approach and applied it to cases in which clusters of controllers are
considered, rather than fully separate controllers.
In real life instances we realize that many of the assumptions we pose in the introduction of our
paper, under which our approach can guarantee convergence and optimality, are far too restrictive;
in later sections of this paper we show how an implementation of this algorithm performs when
facing problems violating these assumptions. While not fully optimal, a consistent decrease in
objective function can be obtained, fully comparable to centralized approaches.
18
4. Proof of Concept
In this section we present a small case study and solve it analytically, in order to show how the
objective function decomposition described in Section 3 can be applied.
The case study is presented in Figure 4.1. This network represents the smallest example in which
different controllers are responsible over different portions of a traffic network, and only share part
of it between each other.
Figure 4.1: Two-controller network with link labels
This network is composed of three OD pairs A-D, B-E and C-F. All links but 10, 11, 12, and 13 have a
constant cost function cl ( fl )  c0 , while for these a linear cost function is applied, as follows:
Link 10
c10 ( f10 )  a10· f10  b10
Link 11
c11 ( f11 )  a11· f11  b11  g11
Link 12
c12 ( f12 )  a12· f12  b12  g12
Link 13
c13 ( f13 )  a13· f13  b13
Table 1
The control vector is defined as g  [ g11 , g12 ] , pricing set on the two links 11 and 12.
According to our decomposition procedure, the sensitivity matrices would, in this case, take the
following form (for the sake of clarity, we only show values related to the four links with nonconstant cost functions):
19
 f10 / g11

f / g11
J f  g    11
 f12 / g11

 f13 / g11
f10 / g12 

f11 / g12 
f12 / g12 

f13 / g12 
TC  f    TCN / f10 TCN / f11 TCN / f12 TCN / f13 
(29)
(30)
While the elements of J f ( g ) need to be computed based on full knowledge of the network, the
single elements of TC ( f ) can be computed as such:
TCN TCA TCB TCC



fl
fl
fl
fl
(31)
where by the notations TC A , TCB , TCC we mean the parts of the Total Cost objective function
pertaining to the sub-networks that could be, for example, monitored by the three separate ODs A, B
and C. This property, as we discussed earlier, is of considerable importance; it means that the
process of constructing TC ( f ) can be entirely distributed among the different entities, who simply
need sharing the results. In this example we imagine that the three entities A, B and C are
responsible for data acquisition alone. This property is, though, valid only for objective functions fully
separable with respect to flows; as we discussed above, if this property is violated, the different
entities would then need some communication and iterations in this process before a stable
sensitivity measurement is reached.
In this simple scenario, the equilibrium problem can be analytically determined in terms of the
following route flows:
f10  f r1
f11  f r2  f r3
f12  f r4  f r5
(32)
f13  f r6
where the routes r1  r6 are defined as follows:
r1  {l1 , l4 , l10 , l14 , l20 }
r2  {l1 , l5 , l11 , l15 , l20 }
r3  {l2 , l6 , l11 , l16 , l21}
r4  {l2 , l7 , l12 , l17 , l21}
r5  {l3 , l8 , l12 , l18 , l22 }
r6  {l3 , l9 , l13 , l19 , l22 }
20
(33)
The elements of our sensitivity reformulation can be determined algebraically as function of
equilibrium route flows, rather than equilibrium link flows (the full specification of this problem can
be found in Appendix B).
a12 a13
a11a13




a12 a13
a11a13


 (a10 a12  a10 a13 )
a10 a13  a11a13  1
J fr ( g )  
·
  a10 a13  a11a13   D
 a10 a12  a10 a13


a10 a12
a10 a11


a10 a12
a10 a11


(34)
With D  a10 a11a12  a10 a11a13  a10 a12 a13  a11a12 a13
2a10 f r1  b10




 a11 f r2  a11 f r2  f r3  b11  a11 f r3 


 a11 f r2  a11 f r2  f r3  b11  a11 f r3 

TTC  f r   
 a12 f r  a12 f r  f r  b12  a12 f r 
4
4
5
5


a f  a f  f b  a f 
12
r4
r5
12
12 r5
 12 r4



2
a
f

b
13 r6
13










(35)
Finally, in the case of pricing, the sensitivity of link costs to changes in control is null, due to the fact
that these are counteracted by revenues:
0
c ( g )   
0
(36)
Our first order sensitivity approximation can thus be compiled:
TCN  g   J fr
T
 g  TC  f r   [ f r ·c( g)]T
T
(37)
and the optimal values for the control vector g can be obtained by solving the following equation:
g*  g : TC N  g   0
(38)
 1

  2  a10b11  a10b13  a13b10  a13b11  


a10  a13
*


g 
 1

   a10b12  a10b13  a13b10  a13b12  
 2

a10  a13


(39)
resulting in:
21
which can be shown to be exactly equal to the result of minimizing the original, full network Total
Cost function TCN  g  .
22
5. Numerical Case Studies
In this Section we confirm the convergence properties discussed in Section 3, and moreover study
the algorithm’s performance when dealing with non-convex problems. We first introduce the
algorithm implementation, together with the necessary basic information, and then proceed to the
numerical tests.
5.1 Introduction: Dial’s B Algorithm
In order to clarify our implementation, we first need to briefly introduce the reader to a well-known
static deterministic traffic assignment algorithm, featuring implicit path enumeration, developed by
Robert Dial (Dial, 2006), called B algorithm.
The effectiveness of Dial’s B algorithm stems from the fact that it exploits the reduction in
computational effort resulting from subdividing the full network equilibrium into sub-equilibrium
problems posed in well-built network partitions, referred to as bushes:
Def: A bush is a subset of arcs of the original problem’s network and comprises an acyclic subnetwork rooted at a given origin, together with the arc flows that carry all and only trips from that
origin to their specific destinations.
The algorithm performs successive equilibrations in the separate origin rooted bushes over
iterations; in his paper, Dial proves that within this procedure any step resulting into an equilibrated,
separate bush also yields a decrease in the full network’s gap function. Moreover, the different
bushes’ topology may change during iterations while seeking a more efficient implicit route set.
We introduce a simple sketch of the algorithm’s procedural steps, in order to help the reader relate
these steps with those of our own algorithm.
To achieve a network-wide, user optimal traffic assignment, Algorithm B proceeds as
follows:



Initialization. For each origin, create its initial bush and feasible arc
flows.
General step. For each origin, transform its current feasible bush into an
equilibrated bush:
i.
Build max- and min-path trees: Find the cheapest path and costliest
used path from the origin to each of the other nodes.
ii.
Equilibrate bush: Shift trips from max- to min-paths to make their
path-cost difference minimal.
iii.
Improve bush: If the bush is not optimal, create a topologically
improved feasible bush with feasible arc flows but containing some
cheaper path(s).
iv.
Reiterate. if (iii) changed the bush, go back to (i) with a new
bush for the same origin; otherwise, continue on at (i) with next
origin’s current bush.
Termination. Quit when every origin’s bush is optimal.
23
While referring the interested reader to Chapter 2 of (Dial, 2006) for further details on the separate
steps, we now comment the rationale behind choosing this specific algorithm as a base for our own
implementation: we chose to extend Dial’s algorithm since it performs an OD-wise decomposition of
the network, when creating bushes. Given the objective of developing distributed optimal control
approaches and the nature of the objective function reformulation introduced in Section 3.1 of this
paper, we consider that extending an algorithm that, by its own nature, already operates based upon
a network-wise separation, allows us to easily obtain all the relevant properties necessary to validate
our ideas.
5.2 The BCD algorithm
We now present an implementation of Algorithm 3.1, which we refer to as BCD Algorithm, short for
“B with Control Distribution”. This algorithm is an extension of Dial’s B algorithm (Dial, 2006), in
which not only static user equilibrium is computed, but, alongside, also optimal control settings
(second best pricing and/or green split) are obtained.
Definitions of parameters employed in the following algorithm:
g0
Initial condition vector
k

 Max
k
K Max
Trust region radius at iteration k
k

Equilibrium precision threshold at iteration k
Trust region update threshold
Maximum trust region radius
Iteration counter
Maximum number of iterations
Control variable convergence threshold
Algorithm 5.1: BCD
Choose an initial point g 0 .
Choose an initial value for the trust region radius Δ0 , an update threshold η , a
maximum trust region radius ΔMax and a maximum number of iterations K Max .
Choose an initial value μ0 for the equilibrium precision threshold (discussed in
details later).
Choose a control variable convergence threshold δ.
1. Set 𝑘 ← 0
2. compute 𝒇𝒌,𝟎 (Equilibrium Assignment)
3. repeat until ‖𝒈𝒌 − 𝒈𝒌−𝟏 ‖ < 𝛿 or 𝑘 ≥ 𝐾𝑚𝑎𝑥 :
for each bush B
for each coordinate direction 𝑒𝑖 in 𝒈𝑩 , set of controls in bush B
i.
update 𝐽̂𝑓 (𝒈) around 𝒈𝑩
𝒇𝒌,𝒊−𝟏
𝒌−𝟏,𝒊 and
𝐵
ii.
compute a local approximation for 𝑓𝑙 (𝑔𝑘−1,𝑖
)∀𝑙
𝑙
iii.
compute a local approximation for 𝑐𝑙 (𝑓𝑘−1,𝑖
)∀𝑙 ∈ 𝐵
iv.
compute a local approximation for 𝑐𝑙 (g Bk−1,i )
̂ 𝑩
v.
update ∇̂
𝑇𝐶 (𝒇) and ∇𝑐 (𝒈 ), based on the approximations
24
vi.
vii.
viii.
ix.
x.
𝐵
form the quadratic model 𝑚𝑘 (𝑔𝑘,𝑖
)
̂
̂
(𝒇)
∇ 𝑇𝐶
and ∇𝑐 (𝒈)
𝐵
solve min 𝑚𝑘 (𝑔𝑘,𝑖
+ 𝑝𝑖𝐵 ) 𝑠. 𝑡. ‖𝑝𝑖𝐵 ‖ ≤ Δ𝑘
based
on
𝐽̂𝒇 (𝒈)
𝑝𝑖
update Δ𝑘 , 𝜇𝑘
if 𝑝𝑖 meets satisfactory criteria
𝐵
𝐵
Set 𝑔𝑘,𝑖
← 𝑔𝑘−1,𝑖
+ 𝑝𝑖𝐵
else
𝐵
𝐵
Set 𝑔𝑘,𝑖
← 𝑔𝑘−1,𝑖
𝐵
compute 𝒇𝒌,𝒊 , to a precision threshold 𝜇𝑘 , based on 𝑔𝑘,𝑖
(Equilibrium Assignment)
update bush 𝐵
xi.
end(for)
if 𝒈𝑩 = ∅
equilibrate bush 𝐵
update bush 𝐵
end(for)
4. Set 𝑘 ← 𝑘 + 1
5. end(repeat)
Rather than iterating along the separate coordinate directions in the full control vector g as for the
general formulation of Algorithm 3.1, we introduce a further subdivision of the network, following
B
Dial’s B algorithm, and therefore subdivide the full control vector among the separate bushes g .
We obtain two inner nested loops for BCD; a loop among all origin-based bushes, that takes care of
subdividing the control vector into all those controllers included in the current bush B. If no
controller is present, the current bush is simply equilibrated following Dial’s B approach. The
innermost loop is, instead, the coordinate-wise decomposition introduced previously.
This innermost loop’s implementation differs with respect to the more general algorithm, we
therefore comment its operations:
B
i.
obtain a linear approximation for the Flow-Control Jacobian J f ( gi ) based on central
ii.
differences; while the control being approximated depends solely on the current bush, the
flow response is that of the whole network.
Fetch from J f ( g ) the sensitivity of all link flows in the network with respect to the current
control value g k 1,i .
iii.
iv.
collect the sensitivity of the local bush’s link costs to flows and controls.
update the linear approximations of TC ( f ) and c ( g ) based on ii. and iii.
v.
based on the above sensitivities, compute the quadratic model mk (·) (see Appendix A for
vi.
the detailed formulation).
minimize the quadratic model within the current trust region radius, and obtain a candidate
B
step pi
vii.
the trust region radius  k and the equilibrium convergence threshold  k are updated.
viii.
ix.
the candidate step is either accepted or refused
the network is re-equilibrated, rather than to a fully consistent equilibrium, to a possibly
lesser threshold  k .
x.
the current bush topology is updated, following Dial B’s algorithm.
The main difference between BCD and Algorithm 3.1, other than the subdivision in bushes, lies in
step ix.; here, rather than computing a full equilibrium every time we accept a change in control, we
limit its convergence to an acceptable threshold  k .
25
The idea behind this is that, as the iterations progress and the system gets closer to a stable local
optimum in the objective function, the model mk (·) will perform better and better, allowing for
bigger optimization steps. The threshold for equilibrium will increase accordingly, since even a few
steps of re-equilibration are sufficient to obtain a stable, correct sensitivity response.
As for Algorithm 3.1, the following indicator

 


g m g  p 
TCN ,i g kB,i  TCN ,i g kB,i  piB
mk
B
k ,i
k
B
k ,i
B
i
(40)
is obtained by measuring the punctual value of the objective function TCN  g  around the different
conditions and comparing them with the model’s interpolation, and employed to update k and  k
over the iterations.
5.3 Case Study 1: three bushes network
In this subsection we apply the BCD algorithm to the network we presented earlier in section 4. We
first employ it on a simple, linear cost function scenario bearing quadratic objective function.
We then introduce BPR cost functions to the problem, and evaluate the resulting error due to the
quadratic approximation of the (non convex) objective function. We also show how the choice of the
initial point 𝑔0 is of dire importance by employing two different scenarios. Finally, we also show a
case in which one of the pricing controllers is substituted by a traffic light controller, and the related
change in objective function solution space.
We compare the performance of our BCD Algorithm with a simpler Bi-Level approach, in which the
lower level problem is the network equilibration, solved by the standard Dial’s B algorithm, while the
upper level problem is the minimization of the Total Cost objective function, performed by Matlab’s
Optimization Toolbox’s fmincon function.
Scenario 1:
The network parameters for this scenario are as follows:
Link 10
c10 ( f10 )  0.1· f10  1
Link 11
c11 ( f11 )  0.2· f11  0.2  g11
Link 12
c12 ( f12 )  0.1· f12  0.4  g12
Link 13
c13 ( f13 )  0.1· f13  1
*
The analytically determined values for the control vector are g  [0.4,0.3] .
We set the algorithm’s parameters as follows:
26
g0  [0, 0]
  0.75
0  100
 0  102
  10 3
In Figure 5.1 the descent of our algorithm (thin line), compared with the descent of fmincon (thick
line), is shown.
Figure 5.1: Algorithm convergence in quadratic solution space
The computational times are 280 s for fmincon and 85.6 s for BCD, both algorithms converge to the
correct solution.
Scenario 2:
This
scenario
employs
the
same
network,
with
BPR
l
cl ( fl )  1   l ·( fl / 400) and with parameter set as follows:
Link 10
Link 11
Link 12
Link 13
10  0.5, 10  2
11  0.2, 11  4
12  0.5, 12  4
13  0.4, 13  2
27
cost
functions
in
the
form
As for the previous case, links 11 and 12 are both subject to pricing, and we compute the optimal
second best pricing values for both, comparing the performances of fmincon and BCD.
The results from this comparison are shown in the following Table:
Links 11,
fmincon
Links
BCD
11,
Comp. Time
12: 161 s
Toll Values
g  0.1574
*
11
𝑻𝑪
7.972 ⋅ 103
𝚫𝑻𝑪
g12*  0.1574
12: 97s
g11*  0.1891
7.973 ⋅ 103
+0.01%
g12*  0.1972
It’s worth mentioning that the solution space, even in a scenario as simple as this, is non-convex. Like
any other locally approximate algorithm, convergence to a global optima cannot be guaranteed
under non-convexity conditions, the algorithm will converge to some local minima, whose location
will be depending on both the initial point g 0 and the trust-region size at each step k k .
This is easily demonstrated by Figures 5.2, 5.3, 5.4 and 5.5, which show the different paths followed
by fmincon(thick line) and BCD(thin line) starting from different initial points. Due to the clear nonconvexity of this example, some initial points lead to local optima, both for fmincon and BCD, a
behaviour highlighted especially in Figures 5.4 and 5.5.
28
Figure 5.2: Algorithm convergence in non-convex solution space, 𝒈𝟎 = [𝟎. 𝟓, 𝟎. 𝟓]
Figure 5.3: Algorithm convergence in non-convex solution space, 𝒈𝟎 = [𝟏, 𝟎. 𝟐𝟓]
29
Figure 5.4: Algorithm convergence in non-convex solution space, 𝒈𝟎 = [𝟎. 𝟐𝟓, 𝟏. 𝟒]
Figure 5.5: Algorithm convergence in non-convex solution space, 𝒈𝟎 = [𝟏. 𝟒, 𝟏. 𝟒]
30
This scenario aimed at showcasing how even a very simple network might bear significant nonconvexities in the objective function, depending on the separate link cost functions. Our decomposed
approach was anyway able to correctly descend from the given initial point towards the same local
minimum as that reached by a fully centralized scheme. The slight difference in the final point is
introduced by the quadratic models being minimized at each iteration, but this hardly influences the
solution’s quality in terms of objective function (0.01%).
Scenario 3:
This scenario employs again the same network, but we now substitute the pricing control on link 11
with a traffic light controller on node 7. We model this traffic light controller by employing a modified
BPR cost function for the two incoming links 5 and 6, modeled as follows:
 fl 
c5 ( f5 )  1   5·

 400· 5 
5
 fl 
c6 ( f 6 )  1   6 ·

 400· 6 
6
(41)
where the two new parameters  5 and  6 model the green split that node 7 assigns to either of the
two incoming links. Finally, we assume that no “all-red” phase is included, and we therefore add the
following constraint to the problem:
5 6 1
The remaining link cost functions are again BPR in the form cl ( f l )  1   l ·( f l / 400)
(42)
l
and with
parameter set as follows:
Link 10
Link 5
Link 6
Link 12
Link 13
10  0.5, 10  2
5  0.2, 5  4
 6  0.2, 6  4
12  0.5, 12  4
13  0.4, 13  2
Link 11’s dependence on flow has been removed in this case, in order to allow a fair comparison
between the solution space for the two pricing controller case and that in which one has been
substituted by a traffic light intersection. Link 12 remains a pricing controlled link, with variable g12 .
Following constraint (42), the solution space for this three-dimensional problem can easily be
expressed by showing only one of the two green splits, in this case we choose that of link 5 and show
in Figure 5.6 the resulting descent from the initial vector g0  [0.56,0.44, 2.5] , respectively
controllers  5 ,  6 and g12 .
31
Figure 5.6: Algorithm convergence in non-convex solution space, with pricing and traffic light controls
The computational times for the two algorithms were as follows: 114 s for fmincon and 63 s for BCD.
As for the previous non-convex case, in this instance our decomposed algorithm was able to reach
the same optimum found by the centralized approach. By analyzing the solution space shape of
Figure 5.6 it’s moreover recognizable that for gradually increasing values of g12 choosing an initial
point such that  5  0.47,  6  0.53 will lead either algorithm to a local minimum in g  [0,1, g12 ] .
We finally show the evolution of the trust region parameter  k for this latter simulation in Figure
5.7.
32
Figure 5.7: Evolution of trust region size
k
over consecutive steps.
From the two figures it’s quite clear that while the quadratic model’s fit to the behaviour of Total
Cost with respect to g12 is quite good already from step 10 onwards, the same conclusion cannot be
drawn for  5 . Indeed, the impact of the traffic light controller is considerably less convex, which
means that the quadratic model’s fit is not to be trusted, at least until later in the iterations, when
the operational point is closer to the well behaved minimum area.
This conclusion is evident when looking at Figure 5.8, where the objective function shape around the
operational point of iteration 10 and iteration 30 is shown for both controllers.
33
Figure 5.8: Shape of the Total Cost objective function for different iteration numbers
5.4 Case Study 2: Sioux Falls Network
The second case study we present is devised in order to assess whether our distributed optimization
approach is systematically able to successfully descend to a local minimum with the same properties
as those reached by centralized approaches. The network used for this case study is shown in Figure
5.7; it’s a very well-known benchmark case; while it can’t be regarded as a real-life size network, it’s
widely recognized as a good candidate for algorithmic testing.
34
3
1
2
1
5
2
4
8
3
4
6
7
35
11
9
13 23
10 31
9
25
33
12
36
15
5
26
32
34 40
6
12
16
21
48
10
29
51 49
30
14
42 71
44
72
23
13
74
39
24
15
18 54
50
18
58
56 60
19
45
46 67
22
59
69 65 68
66
55
7
52
57
70
73 76
16
20
17
28 43
41
38
17
22 47
53
37
19
8
24
27
11
14
21
75
61
63
62
20
64
Figure 5.7 (Prepared by Hai Yang and Meng Qiang, Hong Kong University of Science and Technology)
Rather than employing the full Origin/Destination matrix available in literature, for the sake of
computational speed we limit ourselves to a smaller case, in which the O/D relationships are all the
combinations in which the following node set acts both as Origins and Destinations: [1, 2, 7, 10, 13,
20]. The total demand on this network is 30000veh/h.
In order to validate our algorithm, we set up more than 200 different instances of this network,
randomizing the BPR cost function of all links 𝑙 as follows:
rand  2,4


1
 fl 
cl  fl  
 1  0.25  



rand [120, 70] 
 3000 

(43)
The pricing control vector is g  [ g4 , g14 , g28 , g43 ] , placed on the links highlighted in Figure 5.7 who
frequently appear in the shortest paths of the different O/D patterns.
For all scenarios, we set BCD’s parameters as follows:
35
g0  vect ([0.2])
  0.9
0  102
 0  510
· 2
  10 3
While the choice of g 0 is entirely arbitrary, the four other parameters are chosen following a simple
consideration: as the size and complexity of the problem increases, especially regarding the route
choice behaviour, we adjust the parameters in accordance to the lesser degree of faith we put in the
fact that the solution space might be well-behaved. This shows especially in the decreased value of
0 and the increase of  .
To compare the goodness of the solutions found with our algorithm with classic, centralized
approaches, once a random instance has been created, we first let BCD converge to a solution, and
successively feed this in to MATLAB’s general purpose optimizer fmincon, whose objective function is
the fully centralized Total Cost. We deem successful those instances in which, given an initial point
coming from the decomposed BCD algorithm, little to no improvement is found by employing the
centralized approach. The main idea behind this process is that of discovering whether or not BCD
was able to converge to a local minimum, from which fmincon will not be able to escape. The
instances in which fmincon yields a considerably lower objective function value than BCD mark the
latter’s failure to find a suitable descent direction.
We consider BCD’s performance, in terms of objective function reduction, as the baseline, and
compare fmincon’s result in percentages; a value greater than 0% implies that the centralized
approach was able to find a local minimum with an objective function value lower than that found by
BCD.
The results for these test results is shown in Figure 5.8. Figure 5.9 also includes the extra
computational time spent by fmincon in the comparison criteria.
7000
Total Cost value
6500
6000
BCD
5500
fmincon
5000
4500
1
51
101
151
201
251
Test number
Figure 5.8: Performance comparison between fmincon and BCD
36
1200
% Extra computational time
1000
800
600
400
200
0
0
50
100
150
200
250
% Additional obj. fun. improvement
Figure 5.9: Performance and comp. time comparison between fmincon and BCD
This computational time is also expressed in percentage: a value of 100% implies that the amount of
extra time spent by fmincon equals that which was spent by BCD to determine the initial solution.
The high concentration of test results in the neighborhood of 0% clearly shows that our decomposed
algorithm is able to converge to local minima consistent with fully centralized solutions; moreover,
the computational time analysis shows that seeking further improvement through the centralized
approach is, in most cases, considerably expensive with respect to the actual gain to be found.
The few instances that yield values noticeably different from 0% were analyzed in detail; this analysis
showed that the Total Cost objective function can exhibit, in some cases, flat regions for determinate
controllers. The rationale behind this behaviour can be found by considering the nature of
Deterministic User Equilibrium: if, for a given OD pair containing the controller for which the
sensitivity J f ( g ) is being computed, only one (shortest) route is being utilized, the minimum
amount of cost increase needed to trigger the user’s routing response is the cost gap between this
route and its “next best” alternative. Our approach obtains J f ( g ) by employing a simple Central
Differences scheme, for which the amount of positive/negative control disturbance g is fixed; this
might be smaller than the aforementioned cost gap, in which case the equilibrium will simply not
respond to the cost variation. Matlab’s fmincon procedure, instead, employs techniques that
dynamically recompute the cost disturbance used for the finite difference procedure, and might thus
be luckier in reaching the threshold for activating alternative routes. This problem only affects
deterministic assignment procedures; in a stochastic setting, due to the probabilistic nature of the
users’ perceived costs, even a small disturbance will be sufficient to trigger an equilibrium response,
regardless of the actual cost gap between the shortest and the next best alternative routes.
37
These insights will be exploited in the future, in order to identify those cases in which only one route
alternative is being used by a certain OD pair, so that then a correct sensitivity step or, alternatively,
a better initial point can be determined.
6. Conclusions
We study and define how anticipatory network control can be decomposed while maintaining
network-wide optimality, focusing on static scenarios. We define which and how the sensitivity
information needed by each separate controller can be obtained, identifying through a simple
objective function reformulation which parts of this sensitivity analysis depend on network-wide
information and which, instead, can be estimated separately by entities different than the controllers
themselves.
We represent control decompositon by developing a decomposed optimization approach and
algorithm, in which the physical separation is modeled by controller-wise partitioning of the original
centralized optimization problem. This algorithmic definition allows us to determine the conditions
under which a fully decomposed control scheme would yield performance equal to that of
centralized approaches.
Recognizing that the conditions under which our approach would be fully optimal are restrictive, we
perform different tests, aimed at assessing the decomposed algorithm’s convergence properties and
optimality when dealing with highly non-convex solution spaces. We employ a well-known general
purpose optimization algorithm in order to provide comparisons between our decomposed
approach’s performance and that of a fully centralized approach.
Our test results show that, even when dealing with highly non-convex problems, our decomposed
algorithm is still able to consistently converge to good local minima, therefore confirming our
intuition that optimal control decomposition, even in bigger networks, is still feasible and yields
solutions whose quality is fully comparable to other, local sensitivity based centralized optimization
algorithms.
The few instances in which an acceptable descent direction was not found have been analyzed in
detail, and the rationale behind them was identified for future investigation.
Future research directions include modifying our algorithm’s sensitivity analysis in order to achieve
better robustness when dealing with discontinuities in the objective function’s gradient, to achieve
better results when dealing with deterministic user response. Another interesting direction to pursue
is that of achieving global optimality, e.g. by employing meta-heuristics to escape from local minima.
Naturally distributed approaches, such as Particle Swarm Optimization, could be tuned in order to
fully exploit the structure of our decomposed algorithm; it would be worth investigating how such an
approach would compare to one based on a fully centralized optimization problem.
Finally, our main topic for future research is that of extending the results and findings we obtained in
this work to the domain of Dynamic Traffic Assignment, in order to develop mathematically solid
methods and algorithms that accommodate the need for decomposition of real-life traffic control
38
systems and yet achieve performances comparable to those of a fully centralized anticipatory traffic
controller.
7. Acknowledgments
The authors would like to acknowledge the OT/11/068-project and the Vlaamse Overheid, SBO
project IWT-140433 in the programs ‘Richting Morgen’ and ‘Vlaanderen in Actie Pact 2020’ for
financial support and Willem Himpe for his support in understanding and coding the Dial B algorithm.
39
8. Appendix A: quadratic model formulation
Due to its separable nature, the Total Cost objective function can be expressed as a sum of
components over all links. Based on this consideration, we explicitly define the quadratic model
mk (·) for each controller in the network gi , irrespective of the current iteration, in terms of the
following parametric equation:
1
mk ( gi )  ·H ·gi 2  r·gi  k
2
(44)
where the three parameters are obtained by analytical expansion of Eqn (22):
H  lN
cl fl
·
gi gi
2
(45)
2


f c f 
c f
c
c 
r  lL   g 0,i · l · l  l ·  g 0,i · l · l  fl · l  cl   fl · l 

gi fl gi 
fl gi
fl
gi 




f 
c f
c
c
k  TC0  lL   g 0,i · l ·  g 0,i · l · l  fl · l  cl   g 0,i · fl · l

gi 
fl gi
fl
gi





(46)
(47)
Within the trust region constraints of algorithms 3.1 and 5.1, the solution to this quadratic model is
found through simple algebra:
g i*   H / r
40
(48)
9. Appendix B: analytic route flow solution derivation
Given the link cost functions of Table 1 and the route definitions of (33), the route costs are simply
determined through summation as follows:
cr1  a10 f10  b10  4·c0
cr2  a11 f11  b11  4·c0
cr3  a11 f11  b11  4·c0
cr4  a12 f12  b12  4·c0
(49)
cr5  a12 f12  b12  4·c0
cr6  a13 f13  b13  4·c0
the anticipatory traffic control problem subject to user equilibrium can be defined through the
following system of equations:
cr1  cr2  g11

cr3  g11  cr4  g12
c  g  c
12
r6
 r5

 f r1  f r2  DAD
f  f D
r4
BE
 r3
 f r  f r  DCF
6
 5
(50)
where the first three equations represent the user equilibrium response constraints, that is that the
pair of routes serving the three OD’s share equal costs, while the other three are the demand
satisfaction constraints.
The resulting equilibrium route flows can be analytically determined in function of the demands and
the two control values by solving system (50):
41
1
·(a11a13b10  a12 a13b10  DBE a11a12 a13  DCF a11a12 a13  a11a13b12 
D
a11a13 g12  a12 a13b11  a12 a13 g11  a11a12b10  a11a12b13  DAD a11a12 a13 )
f r1 
1
·(a11 DAD a10 a12  a11a13b10  a12 a13b10  DBE a11a12 a13  DCF a11a12 a13  a11a13b12
D
a11a13 g12  a12 a13b11  a12 a13 g11  a11a12b10  a11a12b13  DAD a10 a11a13  DAD a10 a12 a13 )
f r2 
1
·(a11 DAD a10 a12  a11a13b10  a13a10b11  a13a10 g11  DCF a10 a12 a13  DCF a11a12 a13
D
a10 a13b12  a11a13b12  a10 a13 g12  a11a13 g12  a11a12b10  a10 a12b11  a10 a12b13  a10 a12 g11
f r3  
a11a12b13  DAD a10 a11a13  DBE a10 a12 a13  DBE a11a12 a13 )
1
·(a11 DAD a10 a12  a10 DBE a11a12  a11a13b10  a13a10b11  a13a10 g11  DCF a10 a12 a13
D
 DCF a11a12 a13  a10 a13b12  a11a13b12  a10 a13 g12  a11a13 g12  a11a12b10  a10 a12b11
f r4 
a10 a12b13  a10 a12 g11  a11a12b13  DAD a10 a11a13  a13a10 DBE a11 )
1
·(a11 DAD a10 a12  a10 DBE a11a12  a11a12b10  a10 a11b12  a10 a11b13  a10 a11 g12
D
 a10 a12b11  a10 a12b13  a10 a12 g11  a11a12b13  DCF a10 a11a13  DCF a10 a12 a13  DCF a11a12 a13 )
f r5  
1
·(a11 DAD a10 a12  a10 DBE a11a12  a10 DCF a11a12  a11a12b10  a10 a11b11  a10 a11b13
D
 a10 a11 g12  a10 a12b11  a10 a12b13  a10 a12 g11  a11a12b13 )
f r6 
(51)
With D  a10 a11a12  a10 a11a13  a10 a12 a13  a11a12 a13
Based upon the analytic solution (51) the Jacobian matrix of route flows with respect to variations in
the two controllers can be easily compiled by means of partial derivatives:
 f r1

 f r2
 f
r3
J fr ( g )  
 f r
 4
 f r5

 f r6
/ g11
/ g11
/ g11
/ g11
/ g11
/ g11
f r1 / g12 
a12 a13
a11a13

 


f r2 / g12 
a12 a13
a11a13


f r3 / g12   (a10 a12  a10 a13 )
1
a
a

a
a
10 13
11 13


·
  a10 a13  a11a13   D
f r4 / g12   a10 a12  a10 a13


a10 a12
a10 a11
f r5 / g12  


 
a10 a12
a10 a11

f r6 / g12  
With D  a10 a11a12  a10 a11a13  a10 a12 a13  a11a12 a13
42
(52)
Bibliography
Aboudolas, K., Papageorgiou, M., Kosmatopoulos, E., 2009. Store-and-forward based methods for the
signal control problem in large-scale congested urban road networks. Transp. Res. Part C
Emerg. Technol., Selected papers from the Sixth Triennial Symposium on Transportation
Analysis (TRISTAN VI) 17, 163–174. doi:10.1016/j.trc.2008.10.002
Aboudolas, K., Papageorgiou, M., Kouvelas, A., Kosmatopoulos, E., 2010. A rolling-horizon quadraticprogramming approach to the signal control problem in large-scale congested urban road
networks. Transp. Res. Part C Emerg. Technol. 18, 680–694. doi:10.1016/j.trc.2009.06.003
Allsop, R.E., 1974. SOME POSSIBILITIES FOR USING TRAFFIC CONTROL TO INFLUENCE TRIP
DISTRIBUTION AND ROUTE CHOICE. Presented at the Transportation and Traffic Theory,
Proceedings.
Balaji, P.G., Srinivasan, D., 2010. Multi-Agent System in Urban Traffic Signal Control. IEEE Comput.
Intell. Mag. 5, 43–51. doi:10.1109/MCI.2010.938363
Bar-Gera, H., 2010. Traffic assignment by paired alternative segments. Transp. Res. Part B Methodol.
44, 1022–1046. doi:10.1016/j.trb.2009.11.004
Bar-Gera, H., Hellman, F., Patriksson, M., 2013. Computational Precision of Traffic Equilibria
Sensitivities in Automatic Network Design and Road Pricing. Procedia - Soc. Behav. Sci. 80,
41–60. doi:10.1016/j.sbspro.2013.05.005
Boillot, F., Midenet, S., Pierrelée, J.-C., 2006. The real-time urban traffic control system CRONOS:
Algorithm and experiments. Transp. Res. Part C Emerg. Technol. 14, 18–38.
doi:10.1016/j.trc.2006.05.001
Brdys, M., Tajewski, P., 1994. An algorithm for steady-state optimising dual control of uncertain
plants, in: Proc IFAC Workshop on New Trends in Des of Cont Sys. pp. 249–254.
Cantarella, G.E., Improta, G., Sforza, A., 1991. Iterative procedure for equilibrium network traffic
signal setting. Transp. Res. Part Gen. 25, 241–249. doi:10.1016/0191-2607(91)90141-C
Carlson, R.C., Papamichail, I., Papageorgiou, M., 2014. Integrated feedback ramp metering and
mainstream traffic flow control on motorways using variable speed limits. Transp. Res. Part C
Emerg. Technol. 46, 209–221. doi:10.1016/j.trc.2014.05.017
Conn, A.R., Scheinberg, K., Toint, P.L., 1997. Recent progress in unconstrained nonlinear optimization
without derivatives. Math. Program. 79, 397–414. doi:10.1007/BF02614326
De Oliveira, L.B., Camponogara, E., 2010. Multi-agent model predictive control of signaling split in
urban traffic networks. Transp. Res. Part C Emerg. Technol., Information/Communication
Technologies and Travel Behaviour Agents in Traffic and Transportation 18, 120–139.
doi:10.1016/j.trc.2009.04.022
Diakaki, C., Papageorgiou, M., Aboudolas, K., 2002. A multivariable regulator approach to trafficresponsive network-wide signal control. Control Eng. Pract. 10, 183–195. doi:10.1016/S09670661(01)00121-6
Dial, R.B., 2006. A path-based user-equilibrium traffic assignment algorithm that obviates path
storage and enumeration. Transp. Res. Part B Methodol. 40, 917–936.
doi:10.1016/j.trb.2006.02.008
Dial, R.B., 1999. Network-Optimized Road Pricing: Part II: Algorithms and Examples [WWW
Document]. URL http://pubsonline.informs.org/doi/abs/10.1287/opre.47.2.327 (accessed
1.28.14).
Dotoli, M., Fanti, M.P., Meloni, C., 2006. A signal timing plan formulation for urban traffic control.
Control Eng. Pract. 14, 1297–1311. doi:10.1016/j.conengprac.2005.06.013
Du, L., Han, L., Li, X.-Y., 2014. Distributed coordinated in-vehicle online routing using mixed-strategy
congestion game. Transp. Res. Part B Methodol. 67, 1–17. doi:10.1016/j.trb.2014.05.003
Fletcher, R., 2013. Practical Methods of Optimization. John Wiley & Sons.
43
Gartner, N.H., 1983. OPAC: A DEMAND-RESPONSIVE STRATEGY FOR TRAFFIC SIGNAL CONTROL, in:
Transportation Research Record. Presented at the 62nd Annual Meeting of the
Transportation Research Board.
Gartner, N.H., Pooran, F.J., Andrews, C.M., 2001. Implementation of the OPAC adaptive control
strategy in a traffic signal network, in: 2001 IEEE Intelligent Transportation Systems, 2001.
Proceedings. Presented at the 2001 IEEE Intelligent Transportation Systems, 2001.
Proceedings, pp. 195–200. doi:10.1109/ITSC.2001.948655
Gentile, G., Meschini, L., Papola, N., 2004. Fast heuristics for continuous dynamic shortest paths and
all-or-nothing assignment, in: AIRO 2004.
Geroliminis, N., Haddad, J., Ramezani, M., 2013. Optimal Perimeter Control for Two Urban Regions
With Macroscopic Fundamental Diagrams: A Model Predictive Approach. IEEE Trans. Intell.
Transp. Syst. 14, 348–359. doi:10.1109/TITS.2012.2216877
Haddad, J., Ramezani, M., Geroliminis, N., 2013. Cooperative traffic control of a mixed network with
two urban regions and a freeway. Transp. Res. Part B Methodol. 54, 17–36.
doi:10.1016/j.trb.2013.03.007
Hajiahmadi, M., van de Weg, G., Tampère, C., Corthout, R., Hegyi, A., De Schutter, B., Hellendoorn, H.,
2015. Integrated Predictive Control of Freeway Networks Using the Extended Link
Transmission Model. IEEE Trans. Intell. Transp. Syst.
Hegyi, A., De Schutter, B., Hellendoorn, H., 2005. Model predictive control for optimal coordination
of ramp metering and variable speed limits. Transp. Res. Part C Emerg. Technol. 13, 185–209.
doi:10.1016/j.trc.2004.08.001
Henry, J.J., Farges, J.L., Tuffal, J., 1984. THE PRODYN REAL TIME TRAFFIC ALGORITHM. Presented at
the IFACIFIPIFORS CONFERENCE ON CONTROL IN.
Hunt, P.B., Robertson, D.I., Bretherton, R.D., Winton, R.I., 1981. SCOOT - A TRAFFIC RESPONSIVE
METHOD OF COORDINATING SIGNALS. Publ. Transp. Road Res. Lab.
Katwijk, R.T. van, 2008. Multi-agent look-ahead traffic-adaptive control. Netherlands TRAIL Research
School, Delft, The Netherlands.
Kotsialos, A., Papageorgiou, M., 2004. Efficiency and equity properties of freeway network-wide
ramp metering with AMOC. Transp. Res. Part C Emerg. Technol. 12, 401–420.
doi:10.1016/j.trc.2004.07.016
Lin, S., De Schutter, B., Xi, Y., Hellendoorn, H., 2012. Efficient network-wide model-based predictive
control for urban traffic networks. Transp. Res. Part C Emerg. Technol. 24, 122–140.
doi:10.1016/j.trc.2012.02.003
Lin, S., De Schutter, B., Xi, Y., Hellendoorn, H., 2011. Fast Model Predictive Control for Urban Road
Networks via MILP. IEEE Trans. Intell. Transp. Syst. 12, 846–856.
doi:10.1109/TITS.2011.2114652
Lin, X., Tampère, C.M.J., Viti, F., Immers, B., 2014. The Cost of Environmental Constraints in Traffic
Networks: Assessing the Loss of Optimality. Netw. Spat. Econ. 1–21. doi:10.1007/s11067014-9228-8
Little, J.D.C., Kelson, M.D., Gartner, N.M., 1981. MAXBAND: A PROGRAM FOR SETTING SIGNALS ON
ARTERIES AND TRIANGULAR NETWORKS, in: Transportation Research Record. Presented at
the 60th Annual Meeting of the Transportation Research Board.
Lowrie, P.R., 1990. SCATS, SYDNEY CO-ORDINATED ADAPTIVE TRAFFIC SYSTEM : A TRAFFIC
RESPONSIVE METHOD OF CONTROLLING URBAN TRAFFIC.
Luo, Z.Q., Tseng, P., 1992. On the convergence of the coordinate descent method for convex
differentiable minimization. J. Optim. Theory Appl. 72, 7–35. doi:10.1007/BF00939948
Mauro, V., Taranto, C. di, 1990. UTOPIA. Control Comput. Commun. Transp.
Mirchandani, P., Head, L., 2001. A real-time traffic signal control system: architecture, algorithms,
and analysis. Transp. Res. Part C Emerg. Technol. 9, 415–432. doi:10.1016/S0968090X(00)00047-4
Papageorgiou, M., Hadj-Salem, H., Blosseville, J.-M., 1991. ALINEA: A LOCAL FEEDBACK CONTROL
LAW FOR ON-RAMP METERING. Transp. Res. Rec.
44
Papageorgiou, M., Mayr, R., 1982. Optimal decomposition methods applied to motorway traffic
control. Int. J. Control 35, 269–280. doi:10.1080/00207178208922618
Rinaldi, M., Tampère, C., 2015. Dynamic Traffic Control Clustering. Presented at the Annual meeting
of the Transportation Board, Washington, DC.
Rinaldi, M., Tampère, C.M.J., Himpe, W., Holvoet, T., 2013. Centralized and Decomposed Anticipatory
Model Predictive Control for Network-Wide Ramp Metering, in: Proceedings of the 16th
International IEEE Annual Conference on Intelligent Transport Systems (ITSC 2013).
Presented at the ITSC 2013, The Hague, pp. 461–467.
Roberts, P.D., 2000. Broyden derivative approximation in ISOPE optimising and optimal control
algorithms, in: Proceedings of the 11th IFAC Workshop on Control Applications of
Optimization, St Petersburg, Russia. pp. 293–298.
Sen, S., Head, K.L., 1997. Controlled Optimization of Phases at an Intersection. Transp. Sci. 31, 5–17.
doi:10.1287/trsc.31.1.5
Smith, M.J., Ghali, M., 1990. The dynamics of traffic assignment and traffic control: A theoretical
study. Transp. Res. Part B Methodol. 24, 409–422. doi:10.1016/0191-2615(90)90036-X
Suwansirikul, C., Friesz, T.L., Tobin, R.L., 1987. Equilibrium decomposed optimization: a heuristic for
the continuous equilibrium network design problem. Transp. Sci. 21, 254–263.
Taale, H., 2008. Integrated Anticipatory Control of Road Networks. A Game-Theoretical Approach.
Taale, H., Hoogendoorn, S., 2012. Network-wide Traffic Management with Integrated Anticipatory
Control. Presented at the Transportation Research Board 91st Annual Meeting.
Taale, H., Hoogendoorn, S.P., 2013. A Framework for Real-Time Integrated and Anticipatory Traffic
Management, in: Intelligent Transportation Systems-(ITSC), 2013 16th International IEEE
Conference on. IEEE, pp. 449–454.
Tseng, P., Yun, S., 2009. A coordinate gradient descent method for nonsmooth separable
minimization. Math. Program. 117, 387–423. doi:10.1007/s10107-007-0170-0
Ukkusuri, S., Doan, K., Aziz, H.M.A., 2013. A Bi-level Formulation for the Combined Dynamic
Equilibrium based Traffic Signal Control. Procedia - Soc. Behav. Sci. 80, 729–752.
doi:10.1016/j.sbspro.2013.05.039
Van Den Berg, M., Hegyi, A., De Schutter, B., 2007. Integrated traffic control for mixed urban and
freeway networks: A model predictive control approach. Eur. J. Transp. Infrastruct. Res. 7.
Verhoef, E.T., 2002. Second-best congestion pricing in general static transportation networks with
elastic demands. Reg. Sci. Urban Econ. 32, 281–310.
Wallace, C.E., Courage, K.G., Reaves, D.P., Schoene, G.W., Euler, G.W., 1984. TRANSYT-7F USER’S
MANUAL.
Wang, F.-Y., 2005. Agent-based control for networked traffic management systems. IEEE Intell. Syst.
20, 92–96. doi:10.1109/MIS.2005.80
Webster, F.V., 1958. TRAFFIC SIGNAL SETTINGS. Road Res. Lab Tech Pap. UK.
Wongpiromsarn, T., Uthaicharoenpong, T., Wang, Y., Frazzoli, E., Wang, D., 2012. Distributed traffic
signal control for maximum network throughput, in: 2012 15th International IEEE Conference
on Intelligent Transportation Systems (ITSC). Presented at the 2012 15th International IEEE
Conference on Intelligent Transportation Systems (ITSC), pp. 588–595.
doi:10.1109/ITSC.2012.6338817
Wright, J.N.S.J., 1999. Numerical Optimization.
Yang, H., Bell, M.G., 1997. Traffic restraint, road pricing and network equilibrium. Transp. Res. Part B
Methodol. 31, 303–314.
Yang, H., Yagar, S., 1995. Traffic assignment and signal control in saturated road networks. Transp.
Res. Part Policy Pract. 29, 125–139.
Yan, H., Lam, W.H., 1996. Optimal road tolls under conditions of queueing and congestion. Transp.
Res. Part Policy Pract. 30, 319–332.
Zhang, H., Roberts, P.D., 1990. On-line steady-state optimisation of nonlinear constrained processes
with slow dynamics. Trans. Inst. Meas. Control 12, 251–261.
doi:10.1177/014233129001200504
45
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