I'm looking to solve the optimization problem $$ min_{x \in C} ~ f(x), $$ where $C \subset R^n$ is a closed, convex, bounded set and $f : R^n \to R$ a Lipschitz differentiable (nonconvex) function.

In my problem, $C$ is the solution set of a difficult convex optimization problem, so the projection onto $C$ and also a linear minimization oracle are intractable to compute in closed-form, thus projected gradient or Frank-Wolfe methods are not applicable.

However, I can efficiently compute a separating hyperplane between a point $\bar x$ and the set $C$. My question is whether iterations of the type

$$ \bar x^{t+1} = x^t - \alpha_t \nabla f(x^t), $$ $$ x^{t+1} = \text{proj}_H(\bar x^{t+1}), $$ have been analyzed in literature or have hope of converging to a stationary point. Here $\{ \alpha_t \}$ is a suitable vanishing step-size sequence, and $proj_H$ the projection onto a separating half-space to the set $C$ at point $\bar x^{t+1}$.