Consider a standard form convex optimization problem: $\min_x f(x)$ with the convex constraints $g_i(x) \leq 0$ for all $i \in [p],$ and a convex objective function $f.$ If it helps any, $f$ can be assumed monotone increasing/decreasing as well.
Suppose there exists a point $\tilde{x}$ such that $g_i(\tilde{x}) = 0,$ and there is no point $x^\prime$ in the neighborhood of $\tilde{x}$ and in the interior of the feasible set such that $f(x^\prime) \leq f(\tilde{x}).$ Is this sufficient to claim that any optimizer of the problem satisfies the $i$'th inequality constraint at optimality, i.e. does the optimizer $x^\star$ satisfy $g_i(x^*) = 0 necessarily?
My gut says "no,'' but I can't seem to construct a counter example analytically. This doesn't appear to break the KKT conditions, for example. But it also does not seem to be implied by them, either.
For context: Attempting to use a convex optimization problem to find a feasible point of a nonconvex set. Finding a convex superset of the set in question is easy, as is finding an objective function which would guarantee my above-mentioned property.
Thanks in advance!