Multiple solutions of mixed variable optimization by multistart Hooke and Jeeves filter method

Abstract

In this study, we propose a multistart method based on an extended version of the Hooke and Jeeves (HJ) algorithm for computing multiple solutions of mixed variable optimization problems. The inequality and equality constraints of the problem are handled by a filter set methodology. The basic ideas present in the HJ algorithm, namely the exploratory and pattern moves, are extended to consider two objective functions and to handle continuous and integer variables simultaneously. This proposal is integrated into a multistart method as a local search procedure that is repeatedly invoked to converge to different global and non-global optimal solutions starting from randomly generated points. To avoid repeated convergence to previously computed solutions, the concept of region of attraction of an optimizer is implemented. The performance of the new method is tested on benchmark problems. Its effectiveness is emphasized by a comparison with a well-known solver.

Improving Efficiency of a Multistart with Interrupted Hooke-and-Jeeves Filter Search for Solving MINLP Problems

“…The present study is an extension of the work presented in [11] in two aspects. First, the multistart algorithm is modified in a way that randomly generated points in the search space that are sufficiently close to other already generated and used points are discarded.…”

“…Thus, the proposed algorithm relies on the HJ approach, as outlined in [15], for solving a MINLP problem. This is an extension of the work presented in [11,12] in the sense that multiple solutions of problems like (1) are computed using a multistart algorithm that invokes a modified HJ filter based local search. The herein proposed 'Interrupted HJ-filter' local search algorithm is an iterative process that is applied to a sampled point X = (x, y)…”

“…The recent work presented in [6] is a detailed literature review and contains theoretical contributions, algorithmic developments, software implementations and applications for MINLP. The area of nonconvex MINLP has deserved a lot of attention from researchers working with heuristics, like genetic algorithm [7], ant colony [8], evolutionary algorithms [9], pattern search algorithms [10], multistart Hooke-and-Jeeves algorithm [11,12] and differential evolution [13].…”

“…The issue of locating multiple solutions of MINLP problems by using a multistart method [14], the Hooke-and-Jeeves (HJ) algorithm [15,16] for the local search, and a filter set methodology [17] to handle inequality and equality constraints has been addressed in the past (see [11]). The HJ algorithm has been extended to define a special pattern of points spread over the search space for the integer variables and continuous variables separately.…”

“…The condition joins the probability of a point being generated in the area of the search space not yet covered and the probability of finding a new minimizer after the decision of invoking the local search. Second, the HJ filter based algorithm [11] is further explored by incorporating a methodology that goals the interruption of the HJ iterations when an accepted trial iterate enters an -neighborhood of an already computed minimizer.…”