Suppose I wish to solve the following optimization problem:$$\begin{array}{rl}\min_{\mu\in\mathbb{R}^n} &\mu^\top c\\\textrm{subject to} & \mu\in C_1\cap C_2\cap\cdots\cap C_k,\end{array}$$where each $C_i\subseteq\mathbb{R}^n$ is a convex set.  Furthermore, I have access to projection operators$$p_i(x):=\left\{\begin{array}{rl}\arg\min_y & \|y-x\|_2\\\textrm{subject to} & y\in C_i.\end{array}\right.$$Is there an optimization algorithm that wraps around the $p_i(\cdot)$'s that can find $\mu$?Notes:The number of sets $k$ is potentially large.  When I tried to apply standard ADMM tricks to this problem, I ended up needing $O(nk)$ space, which is too much.If I add a regularizer $\varepsilon \|\mu\|_2^2$ to the problem, then it looks like projection and I can use a cyclic method like Dykstra's algorithm. But I really would like to solve this problem without regularization.
Minimizing linear objective on intersection of convex sets
optimization;convex optimization;constrained optimization;nonlinear programming;projection
See this recent paper on an extension of stochastic gradient descent that could be used on your problem:https://arxiv.org/abs/1511.03760You could also apply Dykstra's algorithm (or any other algorithm that does alternating projections on convex sets) by setting a target value for $\mu^{T}c \leq \gamma $ and reducing it once feasibility has been achieved.  