A Geometric Perspective on Polynomially Solvable Convex Maximization

TL;DR

This paper introduces a geometric property called comonotonicity that characterizes when convex maximization problems are polynomially solvable, unifying and improving existing tractability results.

Contribution

It defines comonotonicity as a key structural property, providing a unified framework for polynomial solvability of certain convex maximization problems.

Findings

Fixed-rank convex maximization is polynomially solvable under comonotonicity.

The framework recovers known results like convex matroid maximization and SPCA.

A lifting technique improves complexity bounds for standard comonotone regions.

Abstract

Convex maximization encompasses a broad class of optimization problems and is generally NP-hard, even for low-rank objectives. This paper investigates structural conditions under which convex maximization becomes polynomially solvable. From a geometric perspective, we introduce comonotonicity, a structural property of the feasible region crucial for problem tractability, and establish mathematical characterizations of this property. Under comonotonicity and mild additional assumptions, we develop a unified enumerative framework showing that fixed-rank convex maximization is polynomially solvable. This viewpoint recovers several known tractability results that previously required separate analyses, such as fixed-rank convex matroid maximization and sparse principal component analysis (SPCA). Furthermore, for the more structured class of standard comonotone feasible regions, we refine the analysis via a lifting technique to achieve a square-root improvement in the complexity bound. Finally, applications to SPCA and its variants illustrate the broad applicability and effectiveness of the proposed framework.