Introduction

In the last chapter, we have laid down a procedure to solve an optimization problem by first posing it as a martingale problem (see Section 4.12, Chapter 4), whose solution may lead to a local extremization of the cost functional. The stochastic search is next guided to reach global maximum by random perturbation strategies—coalescence and scrambling—specifically devised for the purpose. To realize a single reliable scheme that satisfies the diverse and conflicting needs of an optimization problem defined in terms of multi-cost functions under prescribed constraints is a tough task [Fonseca and Fleming 1995, Deb 2001]. This chapter addresses precisely this issue and considers some modifications to the skeletal optimization approach considered in the last chapter so as to impart greater flexibility with which the innovation process may be designed in the presence of conflicting demands en route to the detection of the global extremum. The efficiency of the global search basically relies upon the ability of the algorithm to explore the search space whilst preserving some directionality that helps in quickly resolving the nearest extremum. The development of the modified setup, referred to as COMBEO (Change Of Measure Based Evolutionary Optimization), recognizes the near impossibility of a specific optimization scheme performing uniformly well across a large class of problems. Recognition of this fact had earlier [Hart et al. 2005, Vrugt and Robinson 2007] led to a proposal of an evolutionary scheme that simultaneously ran different optimization methods for a given problem with some communications built amongst the updates by the different methods. We herein similarly aim at combining a few of the basic ideas for global search used with some well-known optimization schemes under a single unified framework propped up by a sound probabilistic basis.

In a way to be explained in the sections to follow, the ideas (or their possible generalizations) behind some of the existing optimization methods may sometimes be readily included in the present setting—COMBEO—often by just tweaking the innovation process and attempting to incorporate the best practices of some of the available stochastic search methods.