A variant of the outer approximation method for globally minimizing a class of composite functions

In this paper, we consider a constrained optimization problem whose objective function is a composition of two functions g:Rⁿ-→R^p and f:R^p-→R¹. We show that a variant of the outer approximation method generates a globally ϵ-minimum point of f ○g = f(g(.)) over a convex set after finitely many iterations, if g is convex and f is continuous and coordinatewise increasing. Preliminary experiments indicate that the proposed algorithm is reasonably practical for two types of multiplicative programs if p is less than four.