Global optimization using homotopy with 2-step predictor-corrector method

Abstract

In optimization, most established search methods are local searches. Thus the development of a method that can be relied upon to find global solutions are therefore highly significant. Homotopy Optimization with Perturbations and Ensembles (HOPE) is such a method. In HOPE, a large storage space is required to store the points generated during its execution and subsequently its space and time complexity will become higher which causes the operational cost of HOPE to be expensive. This is the weakness of HOPE. In this study, a new method which is an improvement over HOPE called Homotopy 2-Step Predictor-Corrector Method (HSPM) is proposed. HSPM applies the Intermediate Value Theorem (IVT) coupled with the modified Predictor-Corrector Halley method (PCH) to overcome the weakness of HOPE. In HSPM, subintervals within which the extremizers lie, called 'trusted' intervals are found based on IVT. A random point is selected from the 'trusted' interval as an initial point to a local search. Each 'trusted' interval produces one local solution. Lastly, the global solution is determined from the local solutions accumulated. From the results, HSPM has been very successful as a minimization tool. It is able to cope with various types of functions' landscapes and able to detect more than one global solutions. Furthermore, HSPM can identify all the minimizers regardless of the step sizes used by the homotopy function. Hence, it has a high success rate in getting to a global minimizer compared to HOPE. Complexity analysis is employed to show the improvement achieved by HSPM. Based on the analysis, HSPM successfully managed to reduce the computational burden suffered by HOPE and acts as a good method in solving one dimensional optimization problem. However, to cope with the requirements today it needs to be extended to deal with multivariable functions for its future work.