On convex optimization without convex representation

We consider the convex optimization problem P:min_x {f(x) : x ? K}{{rm {bf P}}:{rm min}_{rm {bf x}} {f({rm {bf x}}),:,{rm {bf x}}in{rm {bf K}}}} where f is convex continuously differentiable, and K ì mathbb Rⁿ{{rm {bf K}}subset{mathbb R}^n} is a compact convex set with representation {x ? mathbb Rⁿ : g_j(x) 3 0, j = 1,?,m}{{{rm {bf x}}in{mathbb R}^n,:,g_j({rm {bf x}})geq0, j = 1,ldots,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 f_m{phi_mu} with parameter μ, associated with P, usually defined for concave functions (g _j ). We then show that any limit point of any sequence (x_m) ì K{({rm {bf x}}_mu)subset{rm {bf K}}} of stationary points of f_m, m? 0{phi_mu, mu to 0} , is a Karush–Kuhn–Tucker point of problem P and a global minimizer of f on K.