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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.