1 Combinatorial Problems in Cooperative Control: Complexity and Scalability Carla Gomes and Bart Selman Cornell University Muri Meeting March 2002
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Combinatorial Problems in Cooperative Control:
Complexity and Scalability
Carla Gomes and Bart Selman
Cornell UniversityMuri MeetingMarch 2002
Combinatorial Problems in Cooperative Control:
Complexity and Scalability
Carla Gomes and Bart Selman
Cornell UniversityMuri MeetingMarch 2002
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We are investigating how to scale up solutionsof the ROBOFLAG Drill focusing on:
- Mixed Integer Program (MIP) formulations- Randomization- Approximation methods- Portfolios of Algorithms- Combining MIP and constraint search
techniques.
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Problem RepresentationProblem Representation
ROBOFLAG Drill Formulation by Raff D’Andrea and Matt Earl.
• Problem is hybrid, combining discrete and continuous components, with multiple constraints.
• Represented as a mixed logical system (MLD) in which the objective is to compute optimal control policies that minimize the total score of the game.
Mathematical Formulation of the Optimization Problem Mixed Integer Linear Program
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Scaling Up Mixed Integer Linear Program Formulations (MILP)
Scaling Up Mixed Integer Linear Program Formulations (MILP)
Standard approach for solving MILP:
Branch and Bound
How can we improve upon Branch and Bound strategies?
Ideas:
Randomization
Different search strategies for node selection
Portfolios of algorithms
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Branch & Bound:Depth First vs. Best bound
Branch & Bound:Depth First vs. Best bound
Critical to performance of Branch & Bound is the way
in which the next node to be expanded is selected.
Standard approach:
Best-bound --- select the node with the best LP bound
Alternative:
Depth-first --- often quickly reaches an integer solution
(may take longer to produce an overall optimal value)
Tradeoffs between these choices depend on underlying
problem stucture (Gomes et al. 2001).
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ROBOFLAG TestbedROBOFLAG Testbed
Depth First search works well.
Problems that could not be solved before with best bound using were solved with depth first.
Current largest problem solved with CPLEX using Depth First Search (8 attackers and 3 defenders):
• Integer variables = 4040
• Continuous variables 400
• Constraints - 13580 constraints
• Time - 244 secs
(Matt Earl 2002)
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Much room for improvement…Much room for improvement…
We are not yet incorporating any randomization
or discrete constraint propagation techniques.
Nor are we yet exploiting parallelism using a
portfolio approach.
Doing so should allow us to solve problems at
least one or two orders of magnitude larger.
(100,000 to 500,000 vars and 1,000,000+
constraints)
Also, we should be able to include more complex constraints.
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Other Formulations for Solving the Control Optimization Problem
Other Formulations for Solving the Control Optimization Problem
Encodings that provide “tighter” relaxations for the LP problem.
Approximate representations using abstractions (“synthesize larger movements / trajecturies”).
Less compact representations may allow for more propagation and scale up better.
Constraint Satisfaction Problem (CSP) formulations. (*)
Hybrid CSP/LP formulation.
Approximations based on LP randomized rounding.
(*)Sat – the satisfiability problem is a particular case of CSP;however, we believe that SAT encodings may not scale up well in this domain.
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Overall the Roboflag control problem provides an
excellent test bed for the development of scalable
techniques for complex optimization.
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Auxiliary SlidesAuxiliary Slides
Background on improvements on branch and
bound using randomization and parallel portfolios.
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Branch & Bound(Randomized)
Branch & Bound(Randomized)
• Solve linear relaxation of MIP
• Branch on the integer variables for which the solution of the LP relaxation is non-integer:
apply a good heuristic (e.g., max infeasibility) for variable selection ( + randomization ) and create two new nodes (floor and ceiling of
the fractional value)
• Once we have found an integer solution, its objective value can be used to prune other nodes, whose relaxations have worse values
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The performance of randomized Branch and
Bound varies dramatically, on the same
instance.
In fact, the run time distributions often exhibit
long tails (Heavy-tailed Distributions)
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Heavy-tailed behavior of Depth-firstHeavy-tailed behavior of Depth-first
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So, how can we take advantage of the high
variability of randomized methods?
- restart strategies
- portfolio strategies
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Algorithm Portfolio DesignAlgorithm Portfolio Design
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MotivationMotivation
The runtime and performance of randomized algorithms can vary dramatically on the same instance and on different instances.
Goal: Improve the performance of different algorithms by combining them into a portfolio to exploit their relative strengths.
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Portfolio of AlgorithmsPortfolio of Algorithms
A portfolio of algorithm is a collection of algorithms and / or copies of the same algorithm running interleaved or on different processors.
Goal: to improve on the performance of the component algorithms in terms of:
expected computational cost“risk” (variance)
Efficient Set or Efficient Frontier: set of portfolios that are best in terms of expected value and risk.
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Depth-first vs. Best-bound(logistics planning)
Depth-first vs. Best-bound(logistics planning)
Number of nodes
Cu
mula
tive F
requ
en
cies
Depth-First
~50%
Best-Bound
~30%
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Depth-First and Best and Bound do not dominate each other overall.
What if we have more than one processors or if we interleave processes on a single
processor?
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Portfolio for heavy-tailed search procedures (2 processors)
Portfolio for heavy-tailed search procedures (2 processors)
0 DF / 2 BB
2 DF / 0 BB
Standard deviation of run time of portfolios
Expect
ed r
un t
ime o
f p
ort
folio
s
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Portfolio for heavy-tailed search procedures (20 processors)
Portfolio for heavy-tailed search procedures (20 processors)
0 DF / 20 BB
20 DF / 0 BB
Standard deviation of run time of portfolios
Exp
ecte
d ru
n tim
e of
por
tfol
ios
The optimal strategy is to run
Depth First on the 20 processors!
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Optimal collective behavior can
emerge from suboptimal individual
behavior.
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A portfolio approach can lead to substantial improvements in the expected cost and risk of stochastic algorithms, especially in the presence of heavy-tailed phenomena.
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