Covariance Local Search for Memetic Frameworks: A Fitness Landscape Analysis Approach

Abstract

The design of each agent composing a Memetic Algorithm (MA) is a delicate task which often requires prior knowledge of the problem to be effective. This paper proposes a method to analyse one feature of the fitness landscape, that is the epistasis, with the aim of designing efficient local search algorithms for Memetic Frameworks. The proposed Analysis of Epistasis performs a sampling of points within the basin of attraction and builds a data set containing those candidate solutions whose objective function value falls below a threshold.The covariance matrix associated with this data set is then calculated. The eigenvectors of this covariance matrix are then computed and used as the reference system for the local search: a change of variables is performed and then the local search is performed on the new variables. The Analysis of Epistasis has been implemented on the three local search algorithms composing a popular MA called Multiple Trajectory Search (MTS). Numerical results show that the three modified local search algorithms outperform their original counterparts.

Teaching Mathematics to Computer Scientists: Reflections and a Case Study

“…When the optimum of a multivariate function is searched, a set of candidate solutions can be interpreted as a multivariate distribution, see [4,8]. If only a distribution of points whose objective function value is below a threshold (in a minimisation problem) are saved in the data set, then this distribution describes the geometry of the optimisation problem, see [31]. Like for the case of the PCA, the diagonalisation of the associated covariance matrix, that is the detection of its eigenvectors provides the optimisation algorithm with a set of preferential search directions to perform the search for the the optimum, see [30].…”

Covariance Pattern Search with Eigenvalue-determined Radii

“…In [25], [28], this logic has been coupled with a FLA. After having set a problem specific threshold thre, several points are sampled within D and their objective function calculated. Those points whose objective function value is such that f (x) < thre are stored in a data structure V:…”

“…This concept is broadly used in other contexts, especially in Data Science, and is closely related to Principal Component Analysis [33]. Furthermore, it was shown in [28] that, if the sampling of points in V describes the geometry of the basins of attraction, the directions of the eigenvectors identify the maximum and minimum directional derivative. Numerical results in [25] and [28] show that CPS consistently outperforms the PS that uses B e .…”

“…More specifically, in [25], [26] the eigenvectors of the calculated covariance matrix are used to build the Pattern Matrix of an implementation of generalised Patter Search (PS) [27]. In [28], the same FLA has been successfully implemented in three LS algorithms composing Multiple Trajectory Search [29].…”

Adaptive Covariance Pattern Search

“…This concept is broadly used in other contexts, especially in Data Science, and is closely related to Principal Component Analysis [9]. Furthermore, it was shown in [15] that, if the sampling of points in V describes the geometry of the basins of attraction, the directions of the eigenvectors identify the maximum and minimum directional derivative. Thus, numerical results in [14] and [15] show that CPS consistently outperforms the standard pattern search.…”