TITLE: Optimization with weaker oracle than projection
QUESTION [5 upvotes]: 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 $\bar x^{t+1}$.

REPLY [2 votes]: I would guess that the method is going to converge (weakly), even with constant stepsizes. Off the top of my head I don't know a precise reference. The method is close in spirit to the "hybrid projection  proximal point method" by Solodov and Svaiter, but you have a gradient step instead of a proximal step.