Simultaneous Generalized Hill-Climbing Algorithms for Addressing Sets of Discrete Optimization Problems

Diane E. Vaughan, Sheldon H. Jacobson, Shane N. Hall, Laura A. McLay
2005 INFORMS journal on computing  
Generalized hill climbing (GHC) algorithms provide a framework for using local search algorithms to address intractable discrete optimization problems. Many well-known local search algorithms can be formulated as GHC algorithms, including simulated annealing, threshold accepting, Monte Carlo search, and pure local search (among others). This dissertation develops a mathematical framework for simultaneously addressing a set of related discrete optimization problems using GHC algorithms. The
more » ... lgorithms. The resulting algorithms, termed simultaneous generalized hill climbing (SGHC) algorithms, can be applied to a wide variety of sets of related discrete optimization problems. The SGHC algorithm probabilistically moves between these discrete optimization problems according to a problem generation probability function. This dissertation establishes that the problem generation probability function is a stochastic process that satisfies the Markov property. Therefore, given a SGHC algorithm, movement between these discrete optimization problems can be modeled as a Markov chain. Sufficient conditions that guarantee that this Markov chain has a uniform stationary probability distribution are presented. Moreover, sufficient conditions are obtained that guarantee that a SGHC algorithm will visit the globally optimal solution over all the problems in a set of related discrete optimization problems. iii Computational results are presented with SGHC algorithms for a set of traveling salesman problems. For comparison purposes, GHC algorithms are also applied individually to each traveling salesman problem. These computational results suggest that optimal/near optimal solutions can often be reached more quickly using a SGHC algorithm. iv Acknowledgements Athens, Ohio, for providing the initial motivation and an interesting military manufacturing problem application of the results developed in this dissertation. I also would like to thank Dr. Richard Nance, Director of the Systems Research Laboratory for his support of the work in this dissertation. v I would like to acknowledge my fellow students who have influenced my research directions, helped me develop my presentation skills and facilitated in developing and maintaining my interest in operations research and mathematics. In particular, I would like to thank Dr. , for making work a fun and intellectual experience. vi Dedication This dissertation is dedicated to several important people in my life who have made this effort possible. First and most importantly, this dissertation is dedicated to my husband Mark and our son Andrew, who cooked, cleaned and smiled to help with its completion and who fill our home with happiness. In addition, this dissertation is dedicated to Denise and Patricia, for their love and support. This dissertation is also dedicated to Jim and Jeanne Atwell who were always there when we needed them. I would also like to dedicate this dissertation to Richard Hardy, whose advice about life and career decisions I took seriously. Finally, I wish to dedicate this dissertation to my parents. I would like to thank the Makelas for lovingly accepting me, my dreams and my ambitions.
doi:10.1287/ijoc.1040.0064 fatcat:ghijw4nyjjcovlqkmhuoyvo46q