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Consider convex optimization problems subject to a large number ofconstraints. We focus on stochastic problems in which the objective takes theform of expected values and the feasible set is the intersection of a largenumber of convex sets. We propose a class of algorithms that perform bothstochastic gradient descent and random feasibility updates simultaneously. Atevery iteration, the algorithms sample a number of projection points onto arandomly selected small subsets of all constraints. Three feasibility updateschemes are considered: averaging over random projected points, projecting ontothe most distant sample, projecting onto a special polyhedral set constructedbased on sample points. We prove the almost sure convergence of thesealgorithms, and analyze the iterates' feasibility error and optimality error,respectively. We provide new convergence rate benchmarks for stochasticfirst-order optimization with many constraints. The rate analysis and numericalexperiments reveal that the algorithm using the polyhedral-set projectionscheme is the most efficient one within known algorithms.

Random Multi-Constraint Projection: Stochastic Gradient Methods for Convex Optimization with Many Constraints