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

2025-07-22

AI Summary

This paper investigates the worst-case complexity of active-set methods for maximizing convex quadratic functions subject to linear constraints. Addressing the open question raised at ICPO 2025—“Are constantly many objective evaluations sufficient?”—we construct, for the first time, a quadratic objective function via a recursively defined deformed product formulation, augmented with geometric projection techniques. This construction forces the algorithm to traverse exponentially many vertices along a parabolic boundary of the feasible region. Crucially, it avoids shortcuts induced by high-dimensional primal edges, thereby establishing the first unconditional exponential lower bound on the number of iterations that relies solely on convex quadratic polynomials. This result significantly improves upon prior lower bounds, which required super-logarithmic-degree polynomials. Our work provides a key advance in the complexity analysis of simplex-type methods under arbitrary pivot rules.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Exponential worst-case iterations for active-set convex quadratic maximization
• Resolves if constant degree suffices for active-set lower bounds
• Novel recursive formulation forces active-set to follow parabolic boundary

Innovation

• Extended formulation using deformed products
• Polygonal approximation preserving parabola vertices
• Quadratic objective enforcing active-set path