On convex optimization without convex representation

We consider the convex optimization problem $P:{\min }_x\{f(x):x\in K\}$ where f is convex continuously differentiable, and $K\subset {ℝ}^n$ is a compact convex set with representation $\{x\in {ℝ}^n:g_j\left(x\right)\ge 0\mathrm{,}j=1\mathrm{,}\dots m\}$ 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 ${\varphi }_{\mu }$ with parameter μ, associated with P, usually defined for concave functions (g j ). We then show that any limit point of any sequence $\left(x_{\mu }\right)\subset K$ of stationary points of ${\varphi }_{\mu },\mu \to 0$ , is a Karush–Kuhn–Tucker point of problem P and a global minimizer of f on K.