Scaling Up Solutions for Combinatorial Problems in Cooperative Control

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

2. We are investigating how to scale up solutions • of the ROBOFLAGDrill focusing on: • - Mixed Integer Program (MIP) formulations • - Randomization • - Approximation methods • - Portfolios of Algorithms • - Combining MIP and constraint search • techniques.

3. Problem 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

4. 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

5. 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).

6. 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)

7. 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.

8. 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.

9. Overall the Roboflag control problem provides an • excellent test bed for the development of scalable • techniques for complex optimization.

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

11. 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 createtwo 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

12. 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)

13. Heavy-tailed behavior of Depth-first

14. So, how can we take advantage of the high • variability of randomized methods? • - restart strategies • - portfolio strategies

15. Algorithm Portfolio Design

16. 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.

17. 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.

18. Best-Bound ~50% Depth-First ~30% Depth-first vs. Best-bound(logistics planning) Cumulative Frequencies Number of nodes

19. 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?

20. 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

21. 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

22. Optimal collective behavior can • emerge from suboptimal individual • behavior.

23. 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.