Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

TL;DR

This paper introduces a single-projection method for solving a broad class of convex optimization problems in Hilbert spaces, extending previous linear programming results to nonlinear and more complex settings.

Contribution

It generalizes single-projection techniques to nonlinear convex constraints and multiple solutions, introducing a new 'sharpness' property for the constraint set.

Findings

Provides a quantitative estimate for the distance needed for a projection to solve the problem.

Extends previous linear programming results to nonlinear convex constraints.

Introduces the 'sharpness' property of the constraint set.

Abstract

In this work, we consider a class of convex optimization problems in a real 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 complementary conditions, and (iii) possess non-linear 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.