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

Eleon Bach, Yann Disser, Sophie Huiberts, Nils Mosis

TL;DR

The paper proves an unconditional exponential lower bound for the active-set method in convex quadratic maximization under linear constraints, independent of pivot rules. It constructs an extended formulation via deformed products that projects to a parabola while preserving all vertices, enabling tight control of the algorithm's path. The main result shows the active-set method can require $2^d-1$ iterations on a polytope with $2d$ facets, improving prior bounds and addressing a key open question about constant-degree objectives. This advances understanding of pivot-rule complexity in linear and convex optimization and motivates further exploration of strongly polynomial behavior and extensions to other objective classes.

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 $ω(\log d)$ 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.

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

TL;DR

The paper proves an unconditional exponential lower bound for the active-set method in convex quadratic maximization under linear constraints, independent of pivot rules. It constructs an extended formulation via deformed products that projects to a parabola while preserving all vertices, enabling tight control of the algorithm's path. The main result shows the active-set method can require iterations on a polytope with facets, improving prior bounds and addressing a key open question about constant-degree objectives. This advances understanding of pivot-rule complexity in linear and convex optimization and motivates further exploration of strongly polynomial behavior and extensions to other objective classes.

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 in dimension , 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.

Paper Structure

This paper contains 7 sections, 11 theorems, 31 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Simplex and active-set method
  3. An unconditional exponential bound for active-set
  4. An extended formulation approximating a parabola
  5. Active-set on the extended parabola
  6. Future research
  7. Final step of Lemma \ref{['lem:Q']}

Key Result

Theorem 1.1

There exists a polytope $\mathcal{P}\! \subseteq {\mathbb{R}}^d$ with $2d$ facets, and a convex quadratic polynomial $f\colon \mathbb{R}^d \to \mathbb{R}$, such that, for some starting vertex, the active-set method needs $2^d-1$ iterations to maximize $f$ over $\mathcal{P}$, irrespective of the pivo

Figures (5)

  • Figure 1: Projection of the intersection of the Ben-Tal--Nemirovski ben2001polyhedral polyhedral approximation of the Leibniz cone $\{(t,x) : \|x\| \le t\}$ with the halfspace $t \leq 1$, projected to the plane $t=1$ (purple). Additional vertices (orange) are not preserved, i.e., not projected onto vertices.
  • Figure 2: Polygons $\mathcal{V}_{M,N}$ and $\mathcal{W}_{M,N}$ for $M=10$, $N=8$.
  • Figure 3: Projection of $\mathcal{Q}_i$ to the plane $(\varphi_i(\boldsymbol{x}),\, \varphi_i'(\boldsymbol{x}))$ for $n = 8$, $d = 2$, $i = 4$.
  • Figure 4: Objective function values along vertices and edges of the projection $\Pi(\mathcal{P})$ in \ref{['QP']} for $n=M=16$, $d=4$. Purple edges are monotonically increasing in objective value, while the orange edge and chords have negative gradients at both endpoints.
  • Figure : ActiveSet($f$, $A$, $\boldsymbol{b}$, $\boldsymbol{x}$)

Theorems & Definitions (19)

  • Theorem 1.1: Theorem \ref{['thm:main_details']}
  • Theorem 1.2: Theorem \ref{['thm:extended parabola']}
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Theorem 3.4 in amenta1999deformed
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 9 more