Projected Subgradient Ascent for Convex Maximization

TL;DR

This paper introduces a projected subgradient ascent method for convex maximization, demonstrating convergence to stationary points and connecting to classical algorithms like the conditional gradient method.

Contribution

It extends subgradient ascent to convex maximization with convergence guarantees and links it to existing algorithms such as the conditional gradient method.

Findings

Projected subgradient ascent converges to first-order stationary points.

Using large step sizes relates the method to the conditional gradient algorithm.

The approach simplifies solving convex maximization problems in Hilbert spaces.

Abstract

We consider the problem of maximizing a convex function over a closed convex set in a real Hilbert space. For linear functions, we show that a single orthogonal projection suffices to obtain an approximate solution. For continuous convex functions over convex sets, we show that projected subgradient ascent converges to a first-order stationary point when using arbitrarily large step sizes. Taking the step sizes to infinity leads to a deterministic variant of the conditional gradient algorithm, and iterated linear optimization as a special case.