On convex optimization without convex representation

We consider the convex optimization problem P : min(x){f (x) : x is an element of K} where f is convex continuously differentiable, and K subset of R(n) is a compact convex set with representation {x is an element of R(n) : g(j) (x) >= 0, j = 1,..., m} for some continuously differentiable functions (g(j)). We discuss the case where the g(j)'s are not all concave (in contrast with convex programming where they all are). In particular, even if the g(j) are not concave, we consider the log- barrier function phi(mu) with parameter mu, associated with P, usually defined for concave functions (g(j)). We then showthat any limit point of any sequence (x(mu)) subset of K of stationary points of phi(mu), mu -> 0, is a Karush- Kuhn- Tucker point of problem P and a global minimizer of f on K.