An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections

TL;DR

This paper introduces an adaptive approximate projection subgradient method for nonsmooth convex optimization that allows infeasible iterates and guarantees convergence under certain conditions, with applications in chance constraints and compressed sensing.

Contribution

It presents a novel subgradient method using adaptive approximate projections that permits infeasible iterates and ensures convergence to optimal solutions.

Findings

Convergence is guaranteed with fixed or dynamic step sizes.

Method reduces computational cost by using approximate projections.

Applicable to chance-constrained optimization and compressed sensing.

Abstract

We propose a new subgradient method for the minimization of nonsmooth convex functions over a convex set. To speed up computations we use adaptive approximate projections only requiring to move within a certain distance of the exact projections (which decreases in the course of the algorithm). In particular, the iterates in our method can be infeasible throughout the whole procedure. Nevertheless, we provide conditions which ensure convergence to an optimal feasible point under suitable assumptions. One convergence result deals with step size sequences that are fixed a priori. Two other results handle dynamic Polyak-type step sizes depending on a lower or upper estimate of the optimal objective function value, respectively. Additionally, we briefly sketch two applications: Optimization with convex chance constraints, and finding the minimum l1-norm solution to an underdetermined linear system, an important problem in Compressed Sensing.