Projected Subgradient Ascent for Convex Maximization
Pedro Felzenszwalb, Heon Lee
TL;DR
The paper develops a projection-based framework for convex maximization, showing that a single orthogonal projection can approximate linear optimization and that projected subgradient ascent converges to first-order stationary points under large step sizes. It unifies linear and convex cases, linking infinite-step PGA to the conditional gradient method and iterated linear optimization, and provides rigorous bounds on suboptimality and convergence. The approach is extended to semidefinite programming relaxations, notably Max-Cut, by projecting onto the elliptope to obtain Nearest Correlation Matrix solutions with provable error bounds. Empirically, projection-based methods yield comparable SDP objectives and significantly faster runtimes than state-of-the-art solvers on large Max-Cut instances, highlighting practical scalability for convex optimization over complex convex sets.
Abstract
We consider the problem of maximizing a convex function over a closed convex set. Classical methods solve such problems using iterative schemes that repeatedly improve a solution. For linear maximization, we show that a single orthogonal projection suffices to obtain an approximate solution. For general 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 size to infinity leads to the conditional gradient algorithm, and iterated linear optimization as a special case. We illustrate numerical experiments using a single projection for linear optimization in the elliptope, reducing the problem to the computation of a nearest correlation matrix.