"Efficient" Subgradient Methods for General Convex Optimization

TL;DR

This paper introduces an efficient subgradient method for convex optimization that avoids projections, maintains feasibility via line-search, and converges with a relative error rate without requiring Lipschitz continuity.

Contribution

It presents a novel subgradient algorithm that ensures feasibility through line-search, operates without Lipschitz continuity, and achieves convergence rates similar to traditional methods.

Findings

Converges to within user-specified error of optimality.

Avoids orthogonal projections, enhancing practicality.

Operates without Lipschitz continuity requirement.

Abstract

A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified error of optimality. Feasibility is maintained with a line-search at each iteration, avoiding the need for orthogonal projections onto the feasible region (the operation that limits practicality of traditional subgradient methods). Lipschitz continuity is not required, yet the algorithm is shown to possess a convergence rate analogous to rates for traditional methods, albeit with error measured relatively, whereas traditionally error has been absolute. The algorithm is derived using an elementary framework that can be utilized to design other such algorithms.