A finitely solvable class of approximating problems

Let P be the following nonlinear programming problem: given m+1 continuously differentiable convex maps f0 (.), f1(.),..., fm(.) from En into E, minimize f0(z) subject to fj(z) ≥ 0, j=1,2, ..., m. A well known approach for solving P consists of embedding P into a family of approximate problems P(α). Given α>0, the problem P(α) is to find a point z such that fj(z)≤0, j=1,2, ..., m and such that for every h in En there exists j in J(z, α), j depending on h, satisfying ⊽fj(z),h≥ 0, with J(z, α) = {jε{1,2, ...., m}|fj(z)+1/α α 0} u {0}. In general P(α) cannot be solved in a finite number of iterations and therefore one is obliged to use antizigzagging schemes of varying complexity. The purpose of this paper is to describe a class C of problems P such that the approximating problems P(α) may be solved in a finite number of steps.