COMPLEXITY MADE SIMPLE*
*
AT A SMALL PRICE
C. G. Cassandras
Boston University
cgc@bu.edu
http://vita.bu.edu/cgc
Christos G. Cassandras
CODES Lab. - Boston University
THREE FUNDAMENTAL COMPLEXITY LIMITS
NP-HARD
NP-HARD
LIMIT
LIMIT
1/2
1/T
1/T1/2
LIMIT
LIMIT
oneorder
orderincrease
increasein
inestimation
estimationaccuracy
accuracyrequires
requires
one
INFO .
two
orders
increase
in
learning
effort
SPACE
two
orders increase
in learning
effort
Tradeoff
between
GENERALITY
and
EFFICIENCY
STRATEGY
DECISION
Tradeoff between
GENERALITY
andT)
EFFICIENCY
(e.g.,
simulationlength
length
T)
(e.g.,
simulation
=
ofan
analgorithm
algorithm
of
SPACE
SPACE
[Wolpert
and
Macready,
1997]
[Wolpert and Macready, 1997]
N
NO
O-F
-FREE
REE-L
-LUNCH
UNCH
LIMIT
LIMIT
Christos G. Cassandras
CODES Lab. - Boston University
THREE FUNDAMENTAL COMPLEXITY LIMITS
NP-HARD
NP-HARD
LIMIT
LIMIT
1/2
1/T
1/T1/2
LIMIT
LIMIT
Effect is
MULTIPLICATIVE!
N
NO
O-F
-FREE
REE-L
-LUNCH
UNCH
LIMIT
LIMIT
Christos G. Cassandras
CODES Lab. - Boston University
CENTRAL THEME OF THIS TALK...
Rather than tackling hopelessly complex
problems by brute force...
• Seek “surrogate” simpler problems whose
solution is the same or “good enough”
SMALLPRICE:
PRICE:NEAR
NEAROPTIMALITY?
OPTIMALITY?
SMALL
• Exploit problem structure whenever possible
SMALLPRICE:
PRICE:SUFFICIENT
SUFFICIENTGENERALITY?
GENERALITY?
SMALL
Christos G. Cassandras
CODES Lab. - Boston University
OUTLINE
ƒ The classic complex optimization problem
ƒ “SURROGATE PROBLEM” METHODS:
Solutions to hard problems can be recovered from those of simpler problems
ƒ CONCURRENT ESTIMATION:
Answering N “what if” questions may not need N trials
ƒ DECOMPOSITION:
Solutions to hard problems can be recovered from those of simpler problems
ƒ APPLICATIONS TO SOME COMPLEX SYSTEMS:
- Resource Allocation
- Hybrid Systems
Christos G. Cassandras
CODES Lab. - Boston University
TYPICAL OPTIMIZATION OF
COMPLEX SYSTEMS: REPEATED TRIAL AND ERROR
SE
ACE
P
S
H
ARC
SIMULATE SYSTEM
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
SIMULATE SYSTEM
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
etc…etc…etc
Christos G. Cassandras
CODES Lab. - Boston University
RESOURCE ALLOCATION PROBLEMS
N
USERS
K
RESOURCES
• USERS request RESOURCES at random points in time
• USERS hold RESOURCES for random periods of time
EXAMPLES:
EXAMPLES:
Buffersallocated
allocatedto
tolinks
linksin
innetwork
networkswitches
switches
••Buffers
Timeslots
slotsallocated
allocatedto
totasks
tasksin
incomputer
computersystems
systems
••Time
Vehiclesallocated
allocatedto
tomissions
missionsin
intransportation
transportationsystems
systems
••Vehicles
Christos G. Cassandras
CODES Lab. - Boston University
PROBLEM FORMULATION
• ALLOCATION VECTOR: r = [r1,K, rN ], ri ∈ {0,1,2,K}
• Determine r* such that
J d ( r ) = min J d ( r )
*
r∈Ad
COST
COSTFUNCTION
FUNCTION
CONSTRAINT
CONSTRAINTSET
SET
EXAMPLE:
⎧⎪
Ad = ⎨r :
⎪⎩
Christos G. Cassandras
⎫⎪
∑ ri = K ⎬⎪ Capacity Constraint
i =1
⎭
N
CODES Lab. - Boston University
WHY ARE THESE PROBLEMS HARD?
( K + N − 1)!
1. Combinatorial complexity: | S| =
K!( N − 1)!
NP-HARD
NP-HARD LIMIT
LIMIT
2. Stochastic complexity:
Jd(r) = E[Ld(r)] is unknown and can only be
obtained through simulation or direct
observation of actual system sample paths
1/2
1/T
1/T1/2 LIMIT
LIMIT
Christos G. Cassandras
CODES Lab. - Boston University
RESOURCE ALLOCATION EXAMPLE
p1
λ
p2
r1
μ1
1
r2
μ2
2
Allocate K buffer slots
(RESOURCES)
over N servers
(USERS)
so as to minimize
BLOCKING
PROBABILITY
pN
rN
μN
N
For N=6, K=24 → Possible allocations =
Christos G. Cassandras
N
subject to
∑ ri = K
i =1
118,755
CODES Lab. - Boston University
“SURROGATE” PROBLEM APPROACH
PROBLEM:
min J d ( r ) = min Eω [ Ld ( r ,ω )]
r ∈Ad
r ∈Ad
SURROGATE PROBLEM:
[
]
min J c ( ρ ) = min Eω Lc ( ρ ,ω )
ρ ∈Ac
ρ ∈Ac
[ρ[ρ1,…,
ρ ], ρ ∈ℜ+
1,…,ρΝΝ], ρι ι∈ℜ
+
CONTINUOUS
CONTINUOUSSET
SET
CONTAINING
CONTAINING AAd
d
Christos G. Cassandras
CODES Lab. - Boston University
“SURROGATE” PROBLEM APPROACH
CONTINUED
SURROGATE
CONTINUOUS SET AC
DISCRETE SET Ad
2. Observe/Simulate and
estimate gradient
ρn -1
H(rn,ωn)
rn
rn = f(ρn)
ρn
3. Update
1. Transform
r∗
ρ∗
ρn+1
Is r* = f(ρ*) ???
Christos G. Cassandras
CODES Lab. - Boston University
IS r* = f(ρ*) ???
ANSWER: YES...
Theorem: Let ρ* minimize Lc(ρ). Then, there exists a discrete
feasible neighbor r*∈NN(ρ*) which minimizes Ld(r) and satisfies
Ld(r*) = Lc(ρ* ).
[Gokbayrak
[Gokbayrakand
andCassandras,
Cassandras,JOTA,
JOTA,2001]
2001]
Christos G. Cassandras
CODES Lab. - Boston University
“SURROGATE” PROBLEM APPROACH
CONTINUED
SOLUTION: Iterative algorithm, n = 0,1,…, with two steps:
ρn +1 = π n +1[ρn − ηn H (rn , ωn )]
1. Update ρn:
PROJECTION
PROJECTIONINTO
INTO
FEASIBLE
SET
FEASIBLE SETAAc
SENSITIVITY
SENSITIVITYESTIMATE
ESTIMATE
FROM
SIMULATION
FROM SIMULATION
AT
ATDISCRETE
DISCRETEALLOCATION
ALLOCATION
c
STEP
STEPSIZE
SIZE
(LEARNING
(LEARNINGRATE)
RATE)
2. Transform surrogate allocation ρn+1 into
ACTUAL feasible allocation:
rn +1 = f n +1 (ρ n +1 ),
U
O
U
Y
O
OY
D
O
D
W
HHOOW THIISS??
H
T
T
E
T
G
E
G
f n +1 : Ac → Ad
SEVERAL
SEVERALPOSSIBLE
POSSIBLEMAPPINGS
MAPPINGS
Christos G. Cassandras
CODES Lab. - Boston University
LEARNING BY TRIAL AND ERROR
Design
Design
Parameters
Parameters
SYSTEM
Performance
Performance
Measures
Measures
OperatingPolicies,
Policies,
Operating
ControlParameters
Parameters
Control
CONVENTIONAL TRIAL-AND-ERROR ANALYSIS
(e.g., simulation)
• Repeatedly change parameters/operating policies
• Test different conditions
• Answer multiple WHAT IF questions
Christos G. Cassandras
CODES Lab. - Boston University
LEARNING THROUGH PERTURBATION ANALYSIS
Design
Design
Parameters
Parameters
SYSTEM
OperatingPolicies,
Policies,
Operating
ControlParameters
Parameters
Control
WHAT
WHATIF…
IF…
Performance
Performance
Measures
Measures
PERTURBATION
PERTURBATION
ANALYSIS
ANALYSIS
• •Parameter
Parameterpp11==aawere
werereplaced
replacedby
bypp11==bb
• •Parameter
Parameterpp22==ccwere
werereplaced
replacedby
bypp22==dd
••
••
Performance
PerformanceMeasures
Measures
under
underallallWHAT
WHATIFIFQuestions
Questions
ANSWERS TO MULTIPLE “WHAT IF” QUESTIONS
AUTOMATICALLY PROVIDED
Christos G. Cassandras
CODES Lab. - Boston University
PERTURBATION ANALYSIS
BUFFER
MACHINE
Part
Arrivals
Part
Departures
x(t)
2
Observed
sample path:
K=2
1
x(t)
x(t+) = 2
⇒ Δx = -1
x(t+) = 0
⇒ Δx = 0
2
Perturbed
sample path:
K=1
1
Δx = 0
Christos G. Cassandras
Δx = -1
Δx = 0
CODES Lab. - Boston University
PERTURBATION ANALYSIS
CONTINUED
PERTURBATIONDYNAMICS
DYNAMICSOBTAINED
OBTAINED
PERTURBATION
FROMOBSERVED
OBSERVEDNOMINAL
NOMINALSAMPLE
SAMPLEPATH
PATHONLY!
ONLY!
FROM
Δx(t+δ;θ,Δθ) = f[Δx(t;θ,Δθ), x(t;θ); θ, Δθ]
Whydoes
doesthis
thiswork?
work?
Why
Because structural knowledge of nominal system dynamics is also used
ƒ Constructability Theory: Conditions under which this is possible
and methods for constructing perturbed sample paths
ΔJ(t+δδ;;θ,
θ,ΔΔθθ))== f[ΔJ(t;
f[ΔJ(t;θ,
θ,ΔΔθθ),),x(t;
x(t;θθ););θ,
θ,ΔΔθθ]]
ƒ Concurrent Estimation: ΔJ(t+
ƒ Perturbation Analysis: Obtaining unbiased, consistent estimators
dJ
for
dθ
Christos G. Cassandras
CODES Lab. - Boston University
CONCURRENT ESTIMATION
SE
ACE
P
S
H
ARC
OBSERVE SYSTEM
CONCURRENT
ESTIMATION
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
D
I
P
G
A
R NI N
R
A
E
L
OBSERVE SYSTEM
CONCURRENT
ESTIMATION
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
Christos G. Cassandras
CODES Lab. - Boston University
RESOURCE ALLOCATION EXAMPLE
p1
λ
p2
r1
μ1
1
r2
μ2
2
Allocate K buffer slots
(RESOURCES)
over N servers
(USERS)
so as to minimize
BLOCKING
PROBABILITY
pN
rN
μN
N
For N=6, K=24 → Possible allocations =
Christos G. Cassandras
N
subject to
∑ ri = K
i =1
118,755
CODES Lab. - Boston University
“SURROGATE METHOD” RESULTS
Performance
0.4
0.35
0.3
Loss Probability
0.25
Convergence in ~10
iterations vs. 118,755
Series1
0.2
0.15
0.1
0.05
146
141
136
131
126
121
116
111
106
101
96
91
86
81
76
71
66
61
56
51
46
41
36
31
26
21
16
11
6
1
0
Iterations
Christos G. Cassandras
CODES Lab. - Boston University
OUTLINE
ƒ The classic complex optimization problem
ƒ “SURROGATE PROBLEM” METHODS:
Solutions to hard problems can be recovered from those of simpler problems
ƒ CONCURRENT ESTIMATION:
Answering N “what if” questions may not need N trials
ƒ DECOMPOSITION:
Solutions to hard problems can be recovered from those of simpler problems
ƒ APPLICATIONS TO SOME COMPLEX SYSTEMS:
- Resource Allocation
- Hybrid Systems
Christos G. Cassandras
CODES Lab. - Boston University
HIERARCHICAL DECOMPOSITION OF
COMPLEX SYSTEMS
TIME
TIMESCALE
SCALE
MODEL
MODEL
Weeks - Months
??????
??????
Minutes - Weeks
DISCRETE-EVENT
DISCRETE-EVENT
PROCESSES
PROCESSES
Diff. Eq’s, LP
Automata, Simulation,
Queueing
msec - Hours
PHYSICAL
PHYSICAL
PROCESSES
PROCESSES
Christos G. Cassandras
CODES Lab. - Boston University
Diff. Eq’s,
Detailed Simulation
EXAMPLES
• POWER SYSTEMS:
Unit Dynamics
→ Operating Condition Changes
→ Power Flows
• MANUFACTURING:
Physical Part Processing
→ Start/Stop Control
→ Strategic Planning
• HYDRODYNAMICS:
Particle Dynamics
→ Navier-Stokes Equations
Christos G. Cassandras
CODES Lab. - Boston University
HYBRID SYSTEMS
What exactly
DISCRETE-EVENT does that mean?
DISCRETE-EVENT
PROCESS
PROCESS
CONTROL
CONTROL
PHYSICAL
PHYSICAL
(TIME-DRIVEN)
(TIME-DRIVEN)
PROCESS
PROCESS
Christos G. Cassandras
CODES Lab. - Boston University
HYBRID SYSTEM FRAMEWORK
z&2 = g 2 ( z2 , u2 , t )
z&1 = g1 ( z1, u1, t )
TIME
x1
x0
x2
x1
x1 = f1 ( x0 , z1, u1, t )
Christos G. Cassandras
x2 = f 2 ( x1, z2 , u2 , t )
CODES Lab. - Boston University
HYBRID SYSTEM FRAMEWORK
CONTINUED
TIME DRIVEN :
z&i = gi ( zi , ui , t )
TIME TIME TIME TIME
x1
x2
x3
TIME
x4
xi+1
xi+1
x0
x1
x2
x3
xi
xi
[Antsaklis
[Antsaklis(Ed.),
(Ed.),Proc.
Proc.ofofIEEE,
IEEE,2000]
2000]
[Branicky
[Branickyetetal.,
al.,IEEE
IEEETrans.
Trans.on
onAC,
AC,1998]
1998]
Christos G. Cassandras
EVENT - DRIVEN :
xi +1 = f i ( xi , zi , ui , t )
CODES Lab. - Boston University
HYBRID SYSTEM FRAMEWORK
CONTINUED
Physical State, z
z&i = gi ( zi , ui , t )
hybrid
…
x1
Switching Times
Christos G. Cassandras
x2
xi
xi+1 = fi(xi,ui,t)
Temporal State, x
SWITCHING
SWITCHINGTIMES
TIMES
HAVE
THEIR
HAVE THEIROWN
OWN
DYNAMICS!
DYNAMICS!
CODES Lab. - Boston University
OPTIMAL CONTROL PROBLEMS
• Get to a desired final physical state zN in minimum time xN,
subject to N-1 switching events
• Minimize deviations from N desired physical states: (zi - qi)2
and
deviations from target desired times:
(xi - τi)2
In general:
N
min ∑ ∫ Li ( zi (t ), xi , ui (t ))dt
u
i =1
xi
xi −1
Physical state
Christos G. Cassandras
z&i = gi ( zi , ui , t )
s.t.
xi+1 = fi(xi,ui,t)
Temporal state
CODES Lab. - Boston University
TIME-DRIVEN AND EVENT-DRIVEN DYNAMICS
Time-Driven Dynamics (STATE = z):
z&(t ) = g (z , u, t )
or:
zk +1 = g k (zk , uk )
Event-Driven Dynamics (STATE = Event Times xk,i):
xk +1,i = max{xk , j + a k , j u k , j }
j∈Γi
Event counter k = 1,2,...
Event index i∈E = {1,…,n}
Christos G. Cassandras
CODES Lab. - Boston University
HYBRID SYSTEMS IN MANUFACTURING
Key questions facing manufacturing
system integrators:
• How to integrate ‘process control’ with ‘operations control’ ?
• How to improve product QUALITY within reasonable TIME ?
PROCESS CONTROL
• Physicists
• Material Scientists
• Chemical Engineers
• ...
Christos G. Cassandras
OPERATIONS CONTROL
• Industrial Engineers, OR
• Schedulers
• Inventory Control
• ...
CODES Lab. - Boston University
HYBRID SYSTEMS IN MANUFACTURING
CONTINUED
Throughout a manuf. process, each part is characterized by
• A PHYSICAL state (e.g., size, temperature, strain)
• A TEMPORAL state (e.g., total time in system, total time to due-date)
PHYSICAL
STATE
Time-driven
Dynamics
z&i = gi ( zi , ui , t )
NEW
PHYSICAL
STATE
OPERATION
OPERATION
TEMPORAL
STATE
Christos G. Cassandras
xk +1,i = max{xk , j + a k , j u k , j }
j∈Γi
Event-driven
Dynamics
NEW
TEMPORAL
STATE
CODES Lab. - Boston University
HYBRID SYSTEMS IN MANUFACTURING
x i = max {x i − 1 , a i }+ s i ( u i )
EVENT-DRIVEN
COMPONENT
Part
Arrivals
Part
Departures
ai, zi(ai)
TIME-DRIVEN
COMPONENT
Christos G. Cassandras
CONTINUED
xi, zi(xi)
ui
z&i (t ) = g (zi , ui , t )
CODES Lab. - Boston University
EXAMPLE
…a
to nd m
t
u
(e.g his s st be
., d tate
pro
e si
ces
red
sed
tem
per
atu
re)
STATE
STATE
rts
a
t
s
ar t
p
y
r
Eve is state
at th
zi(ui)
si(ui)
Christos G. Cassandras
PROCESS
PROCESS TIME
TIME
CODES Lab. - Boston University
HIERARCHICAL DECOMPOSITION
N
min ∑ ∫ Li ( zi (t ), xi , ui (t ))dt
u
Let:
xi
i =1
z&i = gi ( zi , ui , t )
s.t.
xi −1
si = xi - xi-1
xi+1 = fi(xi,ui,t)
Time spent at ith operating
region (mode)
Consider objective functions:
N
min ∑ [φi ( zi , ui , si ) + ψ i ( xi , si )]
u
i =1
Physical process
Christos G. Cassandras
Switching time process
CODES Lab. - Boston University
HIERARCHICAL DECOMPOSITION
CONTINUED
z&i = gi ( zi , ui , t )
N
min ∑ [φi ( zi , ui , si ) + ψ i ( xi , si )]
u
s.t.
xi+1 = fi(xi,ui,t)
i =1
HIGHER
LEVEL
PROBLEM:
LOWER
LEVEL
PROBLEMS:
N
[
min ∑ φi ( si ) + ψ i ( xi , si )
s
i =1
*
min φi ( zi , ui , si )
ui
]
s.t.
xi+1 = fi(xi,si,t)
s.t.
z&i = gi ( zi , ui , t )
FIXED si
Christos G. Cassandras
CODES Lab. - Boston University
LOWER LEVEL PROBLEM
Parameterized by switching times
LQ PROBLEM:
min φi ( zi , ui , si ) = h ( z fi − zdi ) + ∫
1
2
ui
s.t. z&i = azi + bui , zi (0) = ς i
si
1
0 2
2
2
i
ru (t )dt
Penalize final state deviation
STANDARD LQ SOLUTION METHOD:
φi ( si ) = h( z − zdi ) + ∫
*
Christos G. Cassandras
1
2
*
fi
2
si
1
0 2
*
i
ru (t )dt
CODES Lab. - Boston University
HIGHER LEVEL PROBLEM
N
[
min ∑ φi ( si ) + ψ i ( xi )
s
i =1
*
]
s.t. x i = max {x i − 1 , a i }+ s i ( u i )
Given arrival
sequence (INPUT)
Cost of optimal
process control over
interval [0, si]
Processing time
(CONTOLLABLE)
Cost related to
event timing
EXAMPLE :
Christos G. Cassandras
ψ i ( xi ) = ( xi − τ i )
CODES Lab. - Boston University
2
HYBRID CONTROLLER STRUCTURE
f, ψ
Hybrid controller steps:
Event timing
xi+1 = fi(xi,si,t),
i = 1,…,N
• System identification
s*
• Lower-level solution
Higher-level Controller
• Higher-level solution
• Lower-level solution
• Operation...
s*
φ*(s)
Lower-level Controller
u*(s*)
&z = g ( z, u, t ) g, φ
Physical processes
Christos G. Cassandras
CODES Lab. - Boston University
HOW DO WE SOLVE THE HIGHER LEVEL PROBLEM?
N
[
min ∑ φi ( si ) + ψ i ( xi )
s
i =1
*
Even if these are convex,
problem is still NOT convex in s!
]
s.t. x i = max {x i − 1 , a i }+ s i ( u i )
Causes nondifferentiabilities!
1. Even though problem is
NONDIFFERENTIABLE and NONCONVEX,
optimal solution shown to be unique.
[Cassandras
[Cassandrasetetal.,
al.,IEEE
IEEETrans.
Trans.on
onAC,
AC,2001]
2001]
Christos G. Cassandras
CODES Lab. - Boston University
SOLVING THE HIGHER LEVEL PROBLEM
CONTINUED
2. Optimal state trajectory can be decomposed into “blocks”
a1 a2 a3
a4 a5 a6
x1 x2 x3
x4 x5 x6
a7 a8
x7
a9
IME
T
N
I
T
JUS
x8
Each “block” corresponds to a
Constrained Convex Optimization problem
⇒ search over 2N-1 possible Constrained Convex Optimization problems
BUT
algorithms that only need N Constrained Convex Optimization
problems have been developed ⇒ SCALEABILITY
Christos G. Cassandras
CODES Lab. - Boston University
EXAMPLES: INTERACTIVE JAVA APPLETS
See
http://vita.bu.edu/cgc/Hybrid/
Christos G. Cassandras
CODES Lab. - Boston University
SOME OPEN PROBLEMS, RESEARCH AREAS
• Ordinal Optimization
• Generality of “Surrogate” approaches
• Dynamic Optimization of Complex Systems
(beyond Dynamic Programming…)
• Hybrid Systems: Modeling, Verification,
Stochastic Optimization, Computation Methods
• Robustness vs. Optimality in Complex Systems
• What is the right resolution level for modeling
Complex Systems?
Christos G. Cassandras
CODES Lab. - Boston University
ACKNOWLEDGEMENTS
Thanks to several co-workers whose contributions
are reflected in this talk…
… All 13 Former
+ Current PhD Students…
…Colleagues…
Y.C. (Larry) Ho
Wei-Bo Gong
Yorai Wardi
Young Cho
Liyi Dai
Xiren Cao
Ted Djaferis
Doug Looze
Felisa Vazquez-Abad
…and Sponsors…
NSF, AFOSR, AFRL, DARPA, NASA,
EPRI, ARO, UT, ALCOA, NOKIA, GE
Christos G. Cassandras
CODES Lab. - Boston University