Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1646-1678, June 2024.
Abstract. We consider a class of convex optimization problems in a Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those established for the linear programming setting in Nurminski (2015) by considering problems that (i) may have multiple solutions, (ii) do not satisfy strict complementarity conditions, and (iii) possess nonlinear convex constraints. As a by-product of our analysis, we provide a quantitative estimate on the required distance between the infeasible point and the feasible set in order for its projection to be a solution of the problem. Our analysis relies on a “sharpness” property of the constraint set, a new property we introduce here.