An unconditional lower bound for the active-set method in convex quadratic maximization

TL;DR

This paper proves that the active-set method requires exponential iterations in the worst case for convex quadratic maximization with linear constraints, resolving an open question and advancing understanding of its limitations.

Contribution

It establishes a tight exponential lower bound for the active-set method in convex quadratic maximization, using a novel extended formulation and projection technique.

Findings

Active-set method has exponential worst-case complexity.

The result applies to convex quadratic functions with linear constraints.

It resolves an open question about the method's efficiency.

Abstract

We prove that the active-set method needs an exponential number of iterations in the worst-case to maximize a convex quadratic function subject to linear constraints, regardless of the pivot rule used. This substantially improves over the best previously known lower bound [IPCO 2025], which needs objective functions of polynomial degrees $\omega(\log d)$ in dimension $d$, to a bound using a convex polynomial of degree 2. In particular, our result firmly resolves the open question [IPCO 2025] of whether a constant degree suffices, and it represents significant progress towards linear objectives, where the active-set method coincides with the simplex method and a lower bound for all pivot rules would constitute a major breakthrough. Our result is based on a novel extended formulation, recursively constructed using deformed products. Its key feature is that it projects onto a polygonal approximation of a parabola while preserving all of its exponentially many vertices. We define a quadratic objective that forces the active-set method to follow the parabolic boundary of this projection, without allowing any shortcuts along chords corresponding to edges of its full-dimensional preimage.