Analysis of Random Restart and Iterated Improvement for Global Optimization with Application to the Traveling Salesman Problem

F. Mendivil, R. Shonkwiler and M. C. Spruill
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F. Mendivil: Acadia University
R. Shonkwiler: Georgia Institute of Technology
M. C. Spruill: Georgia Institute of Technology

Journal of Optimization Theory and Applications, 2005, vol. 124, issue 2, No 8, 407-433

Abstract: Abstract The optimization method employing iterated improvement with random restart (I2R2) is studied. Associated with each instance of an I2R2 search is a fundamental polynomial, $$f(x) - o_{0}x + p_{1}x^{2} + \cdots + p_{d}x^{d+1} - 1,$$ in which the coefficient p k is the probability of starting a search k improvement steps from a local minimum. The positive root η of f can be used to calculate the convergence and speedup properties of that instance. Since the coefficients of f are naturally related to the search, it is possible to estimate them online if an a priori estimate of the size θ of the goal basin is available, for example by analysis or prior experience. In this case, the runtime statistical estimate of η converges many times faster than the estimates of the coefficients themselves. The foregoing is illustrated with an application to the traveling salesman problem (TSP) using the k-change as the improvement discipline. Among other things, it is shown that a k-change improvement can be affected by k 2-changes, that θ =1 for convex city sets, and that good estimates of θ can be made from a reduced TSP related to the given one.

Keywords: Random restart; iterated improvement; global optimization; traveling salesman problem (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/s10957-004-0943-z

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