Combinatorial Problems in Cooperative Control: Complexity and Scalability Carla Gomes and Bart Selma

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Combinatorial Problems in Cooperative Control
Complexity and Scalability Carla Gomes and Bart
SelmanCornell UniversityMuri MeetingMarch
2002

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• We are investigating how to scale up solutions
• of 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 Representation

• ROBOFLAG Drill
• Formulation by Raff DAndrea 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)

• 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 BoundDepth 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 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

• 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

• 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 Slides

• Background on improvements on branch and
• bound using randomization and parallel portfolios.

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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-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 Design

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Motivation

• 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 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)
Cumulative Frequencies
Number of nodes
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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)
2 DF / 0 BB
Expected run time of portfolios
0 DF / 2 BB
Standard deviation of run time of portfolios
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Portfolio for heavy-tailed search procedures (20
processors)
0 DF / 20 BB
The optimal strategy is to run Depth First on
the 20 processors!
Expected run time of portfolios
20 DF / 0 BB
Standard deviation of run time of portfolios
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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.