Abstract
This paper presents a framework for analyzing and comparing sub-optimal performance of local search algorithms for hard discrete optimization problems. The β-acceptable solution probability is introduced that captures how effectively an algorithm has performed to date and how effectively an algorithm can be expected to perform in the future. Using this probability, the necessary conditions for a local search algorithm to converge in probability to β-acceptable solutions are derived. To evaluate and compare the effectiveness of local search algorithms, two estimators for the expected number of iterations to visit a β-acceptable solution are obtained. Computational experiments are reported with simulated annealing and tabu search applied to four small traveling salesman problem instances, and the Lin-Kernighan-Helsgaun algorithm applied to eight medium to large traveling salesman problem instances (all with known optimal solutions), to illustrate the application of these estimators.
Similar content being viewed by others
References
Aarts, E., Lenstra, J.K.: Local Search in Combinatorial Optimization. Wiley, New York (1997)
Billingsley, P.: Probability and Measure. Wiley, New York (1979)
Croes, G.A.: A method for solving traveling-salesman problems. Oper. Res. 6, 791–812 (1958)
Dueck, G., Scheuer, T.: Threshold accepting: a general purpose optimization algorithm appearing superior to simulated annealing. J. Comput. Phys. 90(1), 161–175 (1990)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Glover, F., Kochenberger, G.A.: Handbook of Metaheuristics. Kluwer Academic, Boston (2003)
Glover, F., Laguna, M.: Tabu Search. Kluwer Academic, Norwell (1997)
Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman problem heuristic. Eur. J. Oper. Res. 126, 106–130 (2000)
Henderson, D., Jacobson, S.H., Johnson, A.W.: The theory and practice of simulated annealing. In: Glover, F., Kochenberger, G. (eds.) State-of-the-Art Handbook in Metaheuristics, Chap. 10, pp. 287–319. Kluwer Academic, Norwell (2003)
Hogg, R.V., Craig, A.T.: Introduction to Mathematical Statistics. Prentice Hall, Englewood Cliffs (1995)
Hoos, H.H., Stützle, T.: Stochastic Local Search: Foundations and Applications. Elsevier/Morgan Kaufmann, San Francisco (2004)
Jacobson, S.H., Yucesan, E.: Global optimization performance measures for generalized hill climbing algorithms. J. Glob. Optim. 29, 177–193 (2004)
Jacobson, S.H., Yucesan, E.: Analyzing the performance of generalized hill climbing algorithms. J. Heuristics 10(4), 387–405 (2004)
Jacobson, S.H., Sullivan, K.A., Johnson, A.W.: Discrete manufacturing process design optimization using computer simulation and generalized hill climbing algorithms. Eng. Optim. 31, 247–260 (1998)
Johnson, A.W., Jacobson, S.H.: A class of convergent generalized hill climbing algorithms. Appl. Math. Comput. 125, 359–373 (2002)
Johnson, A.W., Jacobson, S.H.: On the convergence of generalized hill climbing algorithms. Discrete Appl. Math. 119, 37–57 (2002)
Knox, J.: Tabu search performance on the symmetric traveling salesman problem. Comput. Oper. Res. 21(8), 867–876 (1994)
Law, A.M., Kelton, W.D.: Simulation Modeling and Analysis, 3rd edn. McGraw Hill, New York (2000)
Lawler, E.L., Lenstra, L.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: The Traveling Salesman Problem. Wiley, Chichester (1985)
Lin, S., Kernighan, B.W.: An effective heuristic for the traveling salesman problem. Oper. Res. 21, 498–516 (1973)
LKH, 2005. LKH Version 1.3 (July 2002). Retrieved Nov 7, 2005 from http://www.akira.ruc.dk/~keld/research/LKH
Nikolaev, A.G., Jacobson, S.H.: Using Markov chains to analyze the effectiveness of local search algorithms. Technical Report, The University of Illinois at Urbana-Champaign, Urbana, IL (2009)
Orosz, J.E., Jacobson, S.H.: Finite-time performance analysis of static simulated annealing algorithms. Comput. Optim. Appl. 21(1), 21–53 (2002)
Reinelt, G.: TSPLIB—a traveling salesman problem library. ORSA J. Comput. 3(4), 376–385 (1991)
Tsubakitani, S., Evans, J.R.: An empirical study of a new metaheuristic for the traveling salesman problem. Eur. J. Oper. Res. 104(1), 113–128 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nikolaev, A.G., Jacobson, S.H., Hall, S.N. et al. A framework for analyzing sub-optimal performance of local search algorithms. Comput Optim Appl 49, 407–433 (2011). https://doi.org/10.1007/s10589-009-9290-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-009-9290-1