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STEEPEST ASCENT DECOMPOSITION METHODS FOR
MATHEMATICAL PROGRAMMING/ECONOMIC EQUILIBRIUM
ENERGY PLANNING MODELS
by
J.F. Shapiro
OR 046-76
February 1976
Supported in part by the U.S. Army Research Office (Durham) under
Contract No. DAHC04-73-C-0032.
ABSTRACT
A number of energy planning models have been proposed for combining
econometric submodels which forecast the supply and demand for energy
commodities with a linear programming submodel which optimizes the
processing and transportation of these commodities. We show how convex
analysis can be used to decompose these planning models into their
econometric and linear programming components. Steepest ascent methods
are given for optimizing the decomposition, or equivalently, for computing
economic equilibria for the planning models.
1.
Introduction
A number of energy planning models have recently been implemented
or proposed which combine (1) econometric submodels for forecasting supply
and demand for energy commodities as functions of the prices on these
commodities with (2) a linear programming submodel for optimizing the
processing and transportation of the
commodities.
Specific models
include, for example, the FEA Project Independence Evaluation System
(Hogan (1974)), the world oil market model of Kennedy (1974), and a
proposed integration of the Brookhaven Energy System Optimization Model
(Hoffman (1973)), with econometric models developed by Data Resources,
Inc. (Jorgenson (1975)).
The models are equilibrium models because
prices, commodities supplied and demanded, and process and transporation
activity levels are all variables to be determined simultaneously in a
generic time period in equilibrium.
The equilibrium conditions can be
interpreted as necessary and sufficient Kuhn-Tucker optimality conditions
for a related concave programming problem which has its own interpretation.
The purpose of this paper is to discuss how mathematical programming
methods can be used to decompose
and solve the concave programming
problem, and thereby the equilibrium model, into its linear programming
and econometric parts.
The linear programming submodel
communicates
with the econometric submodels by passing to them vectors of shadow prices
on the energy commodities.
The shadow prices are optimal for the linear
programming submodelwith fixed commodity levels.
submodels compare
The econometric
the shadow prices with the vector of commodity prices
required to produce the fixed commodity levels assumed in the linear
programming solution.
If these two price vectors are equal, then an
2
equilibrium solution has been reached.
Equivalently, the equilibrium
conditions establish optimality of the prices, commodity levels and
processing and distribution levels in the implied concave programming
problem.
Although we will focus our attention on the analysis and solution
of mathematical programming/economic equilibrium models arising in energy
planning, the approach is appropriate to similar models in other areas.
Included are agriculture models such as the U.S. energy sector model of
Hall et. al. (1975), the world wheat market model of Schmitz and Bawden
(1973), and the water resources planning model of Flinn and Guise (1971).
The plan of this paper is the following.
Section two contains a
statement of the basic concave programming problem to be analyzed,
plus a discussion of how it has been used in energy modeling.
The
following section contains the Kuhn-Tuckeroptimality conditions for the
mathematical programming problem which we interpret as economic equilibrium
conditions.
Two of these optimality conditions cstitute the interface
between econometric forecasting of supply and demand for energy commodities
and optimization of processing and transporting of these commodities.
Section four discusses decomposition methods, based on the optimality
conditions, for computing an optimal solution to the concave
programming problem, or equivalently, for computing an economic equilibrium.
The final section, section five, discusses a number of future areas of
research.
Strict equality is not required between the shadow and commodity price
for a commodity at a zero level.
3
2.
Mathematical Programming/Economic Equilibrium Models
In its mathematical programming form, the basic problem we
wish to analyze and solve is
(la)
+* = max{f(d) - g(s) - cx}
s.t. Ax - s < 0
(lb)
A x - d > O
(lc)
s > O, d > 0, x > 0
(ld)
where f and -g are concave differentiable functions. It is assumed that
(1) has an optimal solution.
The vector d is the demand for energy
commodities and the vector s is the supply of these commodities.
For
reasons that will become clear later, we assume that the inverse
functions -Vf1
and Vg
1
exist on the non-negative orthant.
According
to the inverse function theorem (e.g. Apostol, 1957; p. 144), Vf 1
and Vg 1 will exist on the non-negative orthant if Vf and Vg
have
continuous first partials and non-vanishing Jacobians on that region.
These assumptions appear reasonable for our model.
As we shall see
in the following section, the econometric specification of f and g
will actually be given by Vf
and Vg.
For the moment, the intuitive
justification that f is concave is that the social benefit f(d) due to
satisfied demand d increases monotonically, but at a decreasing rate.
Conversely, the function g is convex because the cost g(s) of delivering
the supply s increases monotonically, but at an increasing rate since
the less expensive quantities are supplied first.
The Project Independence Evaluation System Integrating Model of
the FEA (Hogan (1974)) is a U.S. energy sector model for the year 1985
rery similar to problem (1).
The supply commodities in that model are
coal, oil, gas, synthetics and imports in different regions of the
4
United States.
The commodities demanded are the same physical
co)llmO)d it i'S for in di,it rin
l,
(ornmerr i;l I nn
different regions of the United States.
resit!dent nl use, again in
The FEA model also considers,
at least implicitly, cross cut constraints of the form Bx < b involving
scarce national resources such as steel and capital availability.
The
world oil market model of Kennedy (1974) is an equilibrium model derived
from a problem of the form (1) where the functions f and g are quadratic.
The Brookhaven Energy System Optimization Model (Hoffman (1973)) of the
U.S. energy sector assumes supply and demand in problem (1) are
exogeneously set, and the objective is to minimize the cost of processing
and transportation.
There is a project underway to combine this model
with the interindustry economic model developed by Data Resources, Inc.
(Jorgenson (1975)).
It is hoped that the methodology discussed here
will aid in that integration.
3.
Optimality/Equilibrium Conditions
The interpretation of the Kuhn-Tucker optimality conditions for
a variety of economic models as the embodiment of market equilibrium
conditions has long been recognized (e.g., see Karlin (1959), Intrilligator (1971)).
These models are generally theoretical and the optimality
conditions are used to study existence, uniqueness and stability of the
equilibrium solution.
The difference with the energy planning
models discussed in the previous section is that they are empirical
models consisting of two distinctly different types of submodels which
need to be hooked together; namely, econometric and linear programming
submodels.
In this context, the Kuhn-Tucker optimality conditions
provide a practical mechanism for integrating these diverse models.
Moreover, the purpose of an implemented energy model similar to
(1) is to provide numerical answers.
The optimality conditions are used
in the following section to derive decomposition solution methods for
numerically optimizing problem (1).
Let p and q be vectors of shadow prices on the constraints (lb)
The optimality conditions are:
and (lc), respectively.
The solution
s, d, x is optimal in problem (1) if and only if there exist shadow
prices p, q satisfying
Vg(s) - p > 0
with equality if si > 0
(2a)
Vf(d) - q < 0
with equality if dj > 0
(2b)
-1
-2
c + pA - qA
> 0 with equality if xk > 0
(2c)
1-
(3a)
p(A x - s) = 0
-
2-
-
(3b)
q(A x - d) = 0
1-
-
2-
-
(4a)
A x - s <0
A2x - d > 0
(4b)
s > 0, d > 0, x > 0, p > 0, q > 0
(4c)
The connection between the econometric forecasting submodels
and the linear programming submodel is effected by the conditions (2a)
and (2b).
To see this, let u = Vg(s) and v = Vf(d) denote vectors of
commodity prices on supply and demand, respectively.
Then if s. > 0,
1
condition (2a) states that ui = Pi; that is, the commodity price
for supply commodity i equals the shadow price for that commodity and
they are in equilibrium.
If s
= O, then we permit ui > Pi because a
6
further lowering of the supply price on commodity i would not induce the
supply to increase from 0.
A similar argument holds for the optimality
condition (2b) on the equilibrium between prices on demand commodities
and the relevant shadow prices.
An equilibrium interpretation of the
other optimality conditions is well known and straightforward and is
therefore omitted.
Note, however, that this interpretation does not
depend on the sufficiency of the Kuhn-Tucker conditions due to the concavity of f and -g.
If for some reason these functions were not concave,
then some solutions to the optimality conditions might not be optimal
for problem (1) although they could still be interpreted as equilibrium
solutions.
Thus far we have not considered the computational and empirical
consequences of trying to establish the optimality conditions.
Before
entering into a discussion about solution methods, it is important to
emphasize that typical econometric submodels are designed to compute s
from u and d from v, rather than the inverse relation as we have stated
it in (2a) and (2b).
In other words, the econometric submodels consist
of the functions Vg
and Vf
d = Vf
(v).
which are used to compute s = Vg
(u) and
This implies that in order to hook up the econometric
submodels with the linear programming submodel, we must assume that the
econometric mappings G = Vg
1
and F = Vf
-1
points to give us the values of Vg = G
can be inverted at various
-1
and Vf = F
for use in testing the optimality conditions.
at these points
This might be done
functionally, or by some iterative procedure which exploits the
-1
monotonicity and continuity of Vg -1monotonicity
and Vf
7
4.
Steepest Ascent Decomposition Methods
In this section, we discuss how problem (1) can be solved by
decomposing it into econometric and linear programming submodels.
For s > 0, d > 0, define the function
c(s,d) = f(d) - g(s) + max - cx
Alx < s
s.t.
2
A x > d
x >
It can easily be shown that
(5)
.
is a concave function.
(s,d)
Moreover,
it is continuous, but not everywhere differentiable on the convex
subset of the non-negative orthant where it is finite.
programming duality (ruling out the case that 4(s,d) = +
By linear
since (1) is
assumed to have an optimal solution, but permitting 4(s,d) = -)
4(s,d) = f(d) - g(s) + min ps - qd
s.t. c - pA1 + qA2
>
(6)
p > 0, q > 0.
We assume the convex polyhedral set
i ={(p,q)jc - pAl + qA2 > 0,
is nonempty.
In general,
p > 0, q > 0}
(7)
will be unbounded because we expect there
to be s,d combinations in (5) which do not admit feasible linear programming solutions.
The issue of infeasible s,d combinations could and
8
probably should be handled directly in our subsequent development by
r
r
the generation and use of constraints of the form p s - qrd
rays (pr,qr) of the polyhedron
.
>
0O for
For expositional reasons, however,
we choose to eliminate the possibility that
is unbounded by assuming
that we know a value M > 0 such that all p,q satisfying the optimality
conditions (2), (3), (4) also satisfy
P+
i
M
(8)
j
i
The addition of the constraint (8) to (7) bounds the dual feasible
region and implies that for all s > 0, d > 0,
p(s,d) = f(d) - g(s) +
t
t
min
p s -q d
t=l,...,T
where the (pt,qt) are the dual extreme points.
(9)
Of course, the addition
of the contraint (8) to (6) is equivalent to the addition of an activity
in (5) that permits a feasible linear programming solution to always be
found, but possibly at a very high cost.
The original mathematical programming problem (1) is equivalent
to
'* = max
(sd)
(10)
s.t. s > O, d > 0,
where
(s,d) is given by (9).
The solution of (1) by solving (10) is
a decomposition approach which is illustrated schematically in figure 1.
The computation alternates between the linear programming submodel and
the supply and demand submodels.
A feasible solution s,d,x, to (1) is
generated each time the LP submodel is solved.
As mentioned above, the
9
Fix initial values
s>O, d>O
.!
s,d
.
k4
LP submodel
, .
.
_
s,d, fixed
compute p,q,x
q,d
p,s
-
_
L
if
,I
.
I
_
,
. . .
supply submodel
demand submodel
p,s fixed
q,d fixed
compute y = Vf(d)-q
compute a = p-Vg(s)
.....
.
CT
y
/,y satis
;
optimality
7
Yes - terminate with
optimal s,d,x
conditions
(,
No
-·J
_
,
ascent step
use a,y to
change s,d
.
_
s,d
Figure 1
10
manner of computing a and y in the supply and demand submodels, respectively,
depends upon their structure.
If (o,y) does not satisfy the optimality
conditions for problem (1) (equivalent to and derivable from the optimality
conditions (2), (3), (4)), the s and d in the LP submodel are changed by
taking an ascent step with respect to the function
(s,d).
The steepest
ascent approach to large scale linear programming decomposition is
discussed by Grinold (1972); see also Fisher, Northup and Shapiro (1975).
The algorithmic approaches to be discussed below use the same ideas,
but our main reason for decomposing (1) is to overcome the incompatabilities of
econometric
and linear programming submodels and their realizations as
computer systems.
Steepest ascent
decomposition methods use concepts of convex
analysis which we briefly review.
Rockafellar (1970) gives a thorough
mathematical treatment of convex analysis.
Its relation to decomposition
methods is developed in detail in Shapiro (1976).
of
at (s,d) is a vector satisfying
p(s,d) <
(s,d) +
(s-s) + y(d-d)
If there is a unique subgradient of
of
A subgradient (a,y)
.
for all s,d
(11)
at s,d, then it is the gradient
Any subgradient at (s,d) can be tried as a direction of ascent in
maximizing
(s,d) because it points into the half space containing all
optimal solutions.
The difficulty with this approach is that
may not
acutally increase in a subgradient direction from (s,d) although (s,d)
is not optimal and the function does increase in another subgradient
direction.
11
The difficulty due to multiple subgradients can be overcome by
procedures capable of generating, if necessary, the set af(s,d) of all
subgradients, called the subdifferential.
Define the index set
T(s,d) = {tl(s,d) = f(d) - g(s) + p s - q d}.
Then it can be shown that ai(s,d) is a bounded convex polyhedron with
extreme points (at,yt) = (-Vg(s) + pt, Vf(d) - qt) for some of the
t
T(s,d).
The Kuhn-Tucker optimality conditions (2),
be restated as follows:
(3), (4) can
The solution (s,d) > 0 is optimal in (10)
if and only if there exists
t, t
T(s,d) (equivalently (a,y) e a4(s,d))
satisfying
=
(s)4 t
--i
i
t)T(s,d)
tT(sd)
-0
if d>
1
if s.
< 0
teT(s,d)
Xt -> 0, t
>0
(12)
t
t
if s
O
=
0
=
0
T(s,d)
The optimality conditions (12) for problem (10) are the basis for
solution methods including
(a) subgradient optimization
(b) primal-dual ascent algorithm
(c) simplicial approximation.
12
These methods are not mutually exclusive but complementary, and they
could be integrated, at least conceptually, into a hybrid algorithm.
Space does not permit us to give a great deal of detail about the
application of these methods to (10).
Reference is given to more
detailed treatments of the methods.
(a) subgradient optimization
This is the simplest to implement but it can require considerable
experimentation with parameter settings and could require knowledge
about (10) which we do not have.
It has worked well for nondifferentiable
concave programming problems closely related to (1) the traveling
salesman problem (Held and Karp (1971)) and (2) machine scheduling
problems (Fisher (1976)).
The idea is to generate a sequence of non-negative solutions
{(s ,d )}= to (10) by the rule
k=l
si
= max{si + 0 ai, 01
for all i
(13)
=max{d
d
j
where (
but
¢
J
+
Yj, 01
for all j
,y ) is any subgradient and the scalars
but
e
- 0.
satisfy
0
ncio=l
he
ot tht n atemt i mde o garnte tht 2O.
Note that no attempt is made to guarantee that the function
actually increases from point to point.
Polyak (1967) shows that if
Vg(s ) and Vf(d ) are uniformly bounded, then the (s ,d ) given by (13)
13
will converge to an optimal solution to (10).
The theoretical and
practical rates of convergence may be slow, however,
Thus, Polyak
(1969) suggests the rule
||(
=
4(s' ,
(14)
-
I( ° z
where 0<
< 2 -
l
1 <
ya)jI 2
2 < 2 which has proven superior.
the formula (14) involves knowledge of the maximal value
we do not know, and the functional value
Note that
*,
which
(s ,d ), which we do not
know explicitly but may be able to compute.
Figure 1 is an accurate
description of how subgradient optimization would work on problem (10).
(b)
primal-dual ascent algorithm
This algorithm is given for the piecewise linear case by Fisher
and Shapiro (1974) and Fisher, Northup and Shapiro (1975), and in the
general case by Lemarechal (1974).
algorithm, we must settle for an
any (s,d) > 0 such that
* <
In order to construct a convergent
e-optimal solution (
(s,d) +
.
The algorithm of Lemarechal
(1974) about to be described converges finitely, and
reduced if necessary.
> 0) which is
The algorithm works with
can be successively
E-subgradients of
which are any vectors (a,y) at (s,d) satisfying
~(s,d) <
The set of all
polyhedron.
(s,d) + a(s-s) + y(d-d) +
for all s,d.
-subgradients is denoted by a3 (s,d) and it is a convex
If we let
T (s,d) = {tlf(d) - g(s) + p s - q d
C
<
(s,d) +
l,
14
then the extreme points of
f c(s,d) are included among the points
(ot,yt ) = (-Vg(s) + pt , Vf(d) - q ) for t
(12) with T(s,d) replaced by
for
T(s,d).
The conditions
T (s,d) are necessary and sufficient
-optimality.
The idea of the algorithnis to try at each point (s,d) to
establish the optimality conditions by solving a phase one linear
programming problem.
Since the set T (s,d) can be quite large, the
procedure begins with a small subset.
If the optimality conditions
are not established, then a direction of possible ascent is indicated.
If this direction contains a solution (s',d') such that p(s',d') >
4(s,d) + E, then a step is taken.
Otherwise, the subset of T (s,d)
is augmented by an e-subgradient and the phase one linear programming
is reoptimized.
The primal-dual ascent algorithm has the advantage over
subgradient optimization that it does not require knowledge of
4*,
and the sequential values of ~(s,d) increase by at least e at each
steD.
It has the disadvantage that it does more work at each point
(s,d), and it is more complex to program.
In terms of figure 1, if
the e-subgradient (a,y) does not satisfy the optimality conditions,
then the LP submodel may be resolved, perhaps several times, before
an ascent step is taken.
15
(c)
simplicial approximation
This method has been applied to related types of economic
equilibrium problems by Scarf and Hansen (1973).
In effect, the
method performs a very special type of search over a compact set of non-negative
(s,d) known to contain an optimal solution to (10). The idea is to approximate
(12) by subgradients calculated at distinct, but close together points (s,d).
Space does not permit a fuller development of this method.
details are given by Fisher, Northup and Shapiro
Complete
(1975) for a mathe-
matical programming problem that is sufficiently similar to (10) for
the approach there to be applicable here.
In terms of figure 1, the
simplicial approximation test for termination is the indicated
approximation of the optimality conditions.
If these conditions
are not satisfied, then instead of the ascent step, we have the
exchange of one of the current points in the approximating set for a
new point (s,d) for which a subgradient (,y)
is calculated as shown.
The number of commodities which can be efficiently handled by simplicial
approximation is not yet known.
For the moment, this number appears
to be less than 100, perhaps substantially so.
16
5.
Conclusions and Areas for Future Research
The proposed decompostion scheme for mathematical programming/
economic equilibrium energy planning models is conceptual but fully
implementable.
At the M.I.T. Energy Lab, we are currently considering
an integration of the Brookhaven Energy System Optimization Model
with some of the econometric models developed at M.I.T.
This integration
should provide the ideas given above with a rigorous test.
On the other hand, there remain a number of conceptual questions
to be studied in greater detail.
Hogan (1974) is concerned with the
integrability of the functions Vf and V g, a property which we have
assumed throughout.
This point needs further.
A possibly related
construct which might provide some insight is the Legendre transform
(Rockafellar (1970; chapter 26)) which relates convex properties of
a function to the inverse of its gradient.
An important area of future research is the identification,
analysis and solution of dynamic models derived from (1) whose solutions converge to an optimal solution
to (1).
The econometric supply
aTd demand models are naturally dynamic, and dynamic mathematical
programming submodels can also be constructed (see Shapiro (1975) for
some ideas about how to do this).
In terms of the decomposition approach,
Grinold (1972) gives an ascent algorithm for solving dynamic linear
programming problems as they would arise in this context.
The idea
would be to fix supply and demand levels over the planning horizon,
solve the dynamic linear programming problem, and then adjust the
supply and demand levels in the same spirit as given above.
The
17
dynamic linear programming energy model of Nordhaus (1973) which has
fixed supply and demand levels could be a candidate for this type of
extension.
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