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New Tools for Large-Scale Combinatorial Optimization Problems

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New Tools for Large-Scale Combinatorial Optimization Problems NEW TOOLS FOR LARGE-SCALE COMBINATORIAL OPTIMIZATION PROBLEMS By OLEG V. SHYLO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 °c 2009 Oleg V. Shylo 2 Dedicated to my family 3 ACKNOWLEDGMENTS I would like to thank my adviser, Dr. Panos M. Pardalos, for all of his encouragement and support. His knowledge, experience and sense of humor were crucial for this work. Also, I would like to acknowledge collaborators Clayton Commander, Valeriy Ryabchenko, Alkis Vazacopoulos, Oleg Prokopyev, Ashwin Aruselvan, Ilias Kotsireas and Timothy Middelkoop, who have been a pleasure to work with. I am grateful to my friends, my family and my girlfriend Erika for their love, moral support and all the fun we had together. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 LIST OF TABLES ..................................... 7 LIST OF FIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1 RESTART STRATEGIES IN OPTIMIZATION: PARALLEL AND SERIAL CASES ........................................ 10 1.1 Introduction ................................... 10 1.2 Motivation .................................... 11 1.3 Literature Overview .............................. 12 1.4 Optimal Restart Strategies ........................... 13 1.4.1 Serial Algorithm ............................. 13 1.4.2 Parallel Algorithm ............................ 16 1.5 Algorithm Portfolio and Restart Strategies .................. 20 2 GLOBAL EQUILIBRIUM SEARCH ........................ 25 2.1 Algorithm Description ............................. 25 2.2 Approximation Formula ............................ 29 3 UNCONSTRAINED BINARY QUADRATIC PROBLEM ............ 32 3.1 Introduction ................................... 32 3.2 Description of the Algorithm .......................... 34 3.2.1 Search Procedures ............................ 35 3.2.2 Generation Procedure .......................... 35 3.2.3 Computational Experiments ...................... 35 3.3 Concluding Remarks .............................. 41 4 PERIODIC COMPLEMENTARY BINARY SEQUENCES ............ 42 4.1 Introduction ................................... 42 4.2 PAF and Quadratic Forms ........................... 42 4.3 A Combinatorial Optimization Formalism for PCS .............. 45 4.4 Applications of PCS and Related Sequences ................. 46 4.5 Optimization Algorithm ............................ 47 4.6 Results ...................................... 48 4.6.1 PCS(50, 2) ................................ 48 4.6.2 Consequences .............................. 49 4.7 Conclusion .................................... 49 5 5 WEIGHTED MAX-SAT PROBLEM ........................ 58 5.1 Introduction ................................... 58 5.2 Algorithm for the Weighted MAX-SAT .................... 59 5.2.1 Generation Procedure .......................... 59 5.2.2 Local Search for the Weighted MAX-SAT .............. 59 5.3 Computational Experiments .......................... 61 5.3.1 Benchmarks ............................... 61 5.3.2 Algorithms and Software ........................ 65 5.3.3 Results and Conclusions ........................ 65 6 JOB SHOP SCHEDULING PROBLEM ...................... 67 6.1 Introduction ................................... 67 6.2 Mathematical Model and Notations ...................... 68 6.3 Algorithm Description ............................. 69 6.3.1 Encoding and Generation Procedure .................. 72 6.3.2 Local Search ............................... 74 6.3.3 Acceleration of Move Cost Estimation ................. 76 6.4 Computational Results ............................. 77 6.5 Concluding Remarks .............................. 80 6.6 Benchmark Problems .............................. 82 REFERENCES ....................................... 89 BIOGRAPHICAL SKETCH ................................ 96 6 LIST OF TABLES Table page 3-1 Results for Beasley instances [1] ........................... 39 3-2 Results for Palubeckis’ instances [2] ......................... 40 3-3 Results for the Maximum Independent Set instances arising in coding theory [3] 41 5-1 Results for jnh benchmarks ............................. 64 5-2 Results for rndw1000b benchmarks ......................... 64 5-3 Results for rndw1000c benchmarks ......................... 64 6-1 Computational results by GES for ORB1-ORB10 ................. 82 6-2 Computational results for LA01-LA40 ....................... 83 6-3 Results for TA1-TA80 ................................ 84 6-4 Results for TA1-TA80 (continued) .......................... 85 6-5 Comparison with TSSB ............................... 86 6-6 Comparison with other algorithms ......................... 86 6-7 Results for DMU1-DMU80 .............................. 87 6-8 Results for DMU1-DMU80 (continued) ....................... 88 7 LIST OF FIGURES Figure page 1-1 Tabu algorithm and job shop scheduling problem. Average running time as a function of restart parameter ............................ 11 1-2 The average running time as a function of restart parameter R, ξ has log-normall f(x) distribution with µ = 1 and σ = 2; h(x) = 1−P r{ξ≤x} is hazard rate function ... 15 1-3 The algorithmic speed-up as a function of the number of processors; ξ has log-normal distribution with µ = 1 and σ = 2; the actual speed-up is obtained by comparing the optimal serial restart version to the optimal parallel restart version; the plot for the linear speed-up is for comparison purpposes only. ............. 19 1-4 The efficiency of parallelization (linear speed-up has efficency koefficient equal to 1), ξ has log-normall distribution with µ = 1 and σ = 2; ............ 20 1-5 Example of improved performance when combining two algorithms with log-normally distributed running time. ............................... 21 1-6 Another example of improved performance when combining two algorithms with log-normally distributed running time. ....................... 22 2-1 GES method (general scheme) ............................ 28 3-1 Local Search TABU ................................. 36 3-2 Generation procedure ................................. 37 4-1 Graphical representations of the four symmetric matrices M1,M2,M3,M4 .... 44 5-1 Generation procedure ................................. 60 5-2 Local Search 1-opt .................................. 61 5-3 Local Search k-opt .................................. 62 5-4 Local Search TABU ................................. 63 6-1 GES for JSP ..................................... 70 6-2 Procedure that generates solution ......................... 73 6-3 Improvement procedure ............................... 76 6-4 Estimation procedure ................................ 78 8 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NEW TOOLS FOR LARGE-SCALE COMBINATORIAL OPTIMIZATION PROBLEMS By Oleg V. Shylo August 2009 Chair: Panos M. Pardalos Major: Industrial and Systems Engineering Many traditional algorithmic techniques used in combinatorial optimization have reached the computational limits of their scope. A relatively low cost and availability of parallel multi-core clusters offer a potential opportunity to shift these limits. Unfortunately, most existing serial algorithms are not easily adaptable to parallel computing systems. The objective of this work is to provide new algorithmic methods that can be easily and effectively scaled to parallel systems with large numbers of processing units. We concentrate on a scalability that comes from running a set of independent algorithms in parallel. We investigate the relationship between the parallel acceleration and properties of serial algorithms. Some of these results revealed that the best serial algorithm is not necessarily the best choice in parallel setting. Additionally, we observed that the combination of different algorithms (some of them may have poor performance characteristics!) can be more beneficial then using any particular algorithm on its own. The limitations of this approach are addressed. In this dissertation, we present the algorithms based on Global Equilibrium Search method and Tabu Search method for classical optimization problems. These algorithm demonstrate the state-of-the-art performance comparing to the latest published work in the field: Unconstrained Binary Quadratic Problem, Weighted MAX-SAT problem and Job Shop Scheduling Problem. 9 CHAPTER 1 RESTART STRATEGIES IN OPTIMIZATION: PARALLEL AND SERIAL CASES 1.1 Introduction Let P be a class of problems and A be a randomized algorithm that is able to solve correctly any instance of P in finite time. An algorithm A is of the Las Vegas type, if its running time on a given instance p ∈ P is a random variable tA(p). A restart version of A(p) can be implemented by stopping the algorithm after certain time and running it from the beginning: i.e. run A(p) for time R or until the correct solution is found; restart A(p) from the beginning and run it for time R or until the correct solution is found; and so on. In other words, each run of A(p) is independent from all the others and limited to time R. The current paper investigates the running time distribution, such that the average running time of AR(p) is smaller then the running time of A(p). This type of distributions is ofen observed in practical application, and thus it is crucial to understand the underlying
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