Generating $\alpha $-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization

Abstract

This paper deals with the dimensionality reduction approach to study multi-dimensional constrained global optimization problems where the objective function is non-differentiable over a general compact set D of R n and Hölderian. The fundamental principle is to provide explicitly a parametric representation xi = i(t), 1 ≤ i ≤ n of α-dense curve α in the compact D, for t in an interval I of R, which allows to convert the initial problem to a one-dimensional Hölder unconstrained one. Thus, we can solve the problem by using an efficient algorithm available in the case of functions depending on a single variable. A relation between the parameter α of the curve α and the accuracy of attaining the optimal solution is given. Some concrete α-dense curves in a non-convex feasible region D are constructed. The numerical results show that the proposed approach is efficient.