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(??). Given ??>0, the problem P(??) 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, ??), j depending on h, satisfying ????fj(z),h?? ?? 0, with J(z, ??) = {j??{1,2, ...., m}|fj(z)+1/?? ?? 0} u {0}. In general P(??) 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(??) may be solved in a finite number of steps.