Heuristic Approaches for Non-Convex Problems

Abstract

This paper aims at solving difficult optimization problems arising in many engineering areas. To this end, two recently developed optimization method will be introduced: the heuristic Kalman algorithms (HKA) and the quasi geometric programming (QGP) problems. The principle of HKA is to consider the optimization problem as a measurement process intended to give an estimate of the optimum. A specific procedure, based on the Kalman estimator, is developed to improve the quality of the estimate obtained through a measurement process. A significant advantage of HKA against other stochastic methods lies mainly in the small number of parameters which have to be set by the user. In this paper we also introduce an extension of standard geometric programming (GP) problems which we call quasi geometric programming (QGP) problems. The consideration of this particular kind of nonlinear and possibly non smooth optimization problem is motivated by the fact that many engineering problems can be formulated as a QGP. To solve this kind of problems (QGP), an algorithm is proposed which is based on the resolution of a succession of standard GP. An interesting feature of the proposed approach is that it does not need to develop specific program solver and works well with any existing solver able to solve conventional GP. In the last part of the paper, it is to shown that HKA and QGP can be efficiently used to solve difficult non-convex optimization problems. In particular, we have addressed the problem of robust structured control and on-ship spiral inductor design. Numerical experiments exemplify the resolution of this kind of problems.