# Optimization with weaker oracle than projection

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 $$\bar x^{t+1}$$.