Genetic engineering via negative fitness:Evolutionary dynamics for global optimization

Abstract

In this paper, we provide a novel interpretation of a global optimization procedure forstandard quadratic problems (QPs) in terms of selection models. A standard QP consists ofmaximizing a quadratic form over the standard simplex. Recently, local solutions to standardQPs have been obtained with the help of replicator dynamics, which were designed to modelevolutionary processes. Following trajectories under these dynamics, one obtains a sequenceof feasible points with strictly increasing objective values, which approach stationary points.We present regularity conditions which ensure that the limiting points are indeed localsolutions, so that jamming could be avoided. Genetic engineering via negative fitness is ablock pivoting procedure which either delivers a certificate for global optimality of thecurrent local solution, or else enables an escape from its basin of attraction, if it is inefficient.The name originates from the way the block pivot is obtained: via minimization (rather thanmaximization) of a subproblem. We also elaborate on an important application, the searchfor a maximum clique in an undirected graph, and report both theoretical and empiricalresults.