Abstract
Pattern Search is a family of gradient-free direct search methods for numerical optimisation problems. The characterising feature of pattern search methods is the use of multiple directions spanning the problem domain to sample new candidate solutions. These directions compose a matrix of potential search moves, that is the pattern. Although some fundamental studies theoretically indicate that various directions can be used, the selection of the search directions remains an unaddressed problem. The present article proposes a procedure for selecting the directions that guarantee high convergence/high performance of pattern search. The proposed procedure consists of a fitness landscape analysis to characterise the geometry of the problem by sampling points and selecting those whose objective function values are below a threshold. The eigenvectors of the covariance matrix of this distribution are then used as search directions for the pattern search. Numerical results show that the proposed method systematically outperforms its standard counterpart and is competitive with modern complex direct search and metaheuristic methods.
Highlights
Modern numerical optimisation problems are often complex and do not allow the application of gradient-based algorithms, see [4]
Following the fitness landscape analysis, greedy Covariance Pattern Search (gCPS) quickly improves upon the initial value and reaches a solution whose objective function value is orders of magnitude lower than that detected by the standard gPS
This paper proposes an implementation of Generalised Pattern Search (GPS) in which search directions are given by the eigenvectors of the covariance matrix associated with the distribution of points whose objective function value falls below a prearranged threshold
Summary
Modern numerical optimisation problems are often complex and do not allow the application of gradient-based algorithms, see [4]. Since from a theoretical standpoint, this article does not propose a completely new algorithm but a novel implementation of GPS (that is the use of the basis matrix = ), the convergence of Covariance Pattern Search algorithms is subject to the same hypotheses discussed in [48] and summarised in Theorem 1. Observation 2 The proposed Covariance Pattern Search algorithms converge to a null gradient point (possibly a local optimum) under the same conditions of GPS.
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