Using Extreme Order Statistics of Multimodal Mixture Distributions for Complexity Analysis of Semi-Steady-State Jaya Algorithm

Abstract

Computational complexity analysis of an algorithm is an integral part of understanding and applying that algorithm. For stochastic, adaptive heuristics in non-convex optimization, however, complexity analysis is often difficult. This article derives, for the first time in the literature, the complexity of the semi-steady-state Jaya algorithm (which is a recently developed variant of the Jaya algorithm) without the unimodality assumption. The Jaya algorithm, and its improvement, the semi-steady-state Jaya, are among the newest metaheuristics in population-based, nature-inspired optimization methods. In black-box function optimization, stochastic models of evolutionary and non-evolutionary heuristics often study the search process as sampling from distributions that are difficult to estimate. Unimodal distributions used for this purpose are easy to analyze but are necessarily restrictive. In this article, we model multimodality using mixtures of unimodal densities. For multimodal mixtures of uniform densities and, separately, of exponential densities (with different location parameters for the mixture components), analytical expressions, many of them closed-form, are derived for (i) the expectation of the largest order statistic for samples from the mixture; (ii) asymptotics of the above expectation for the large-sample case; (iii) survival probability corresponding to the (asymptotic) expected value of the largest order statistic; and (iv) asymptotics of sums of survival probabilities. The above quantities are used in a stochastic model of the semi-steady-state Jaya algorithm, obtaining the (asymptotic) expectation of the number of updates of the best individual in a population of the algorithm, which in turn is used in the derivation of the computational complexity of the algorithm.