APX-Hardness of Computing Lipschitz Constants for Multi-Parametric Quadratic Programs

2026-06-02

AI Summary

This study investigates the computational complexity of computing the Lipschitz constant of the solution mapping for multiparametric quadratic programs—a quantity essential for optimization-based control analysis. By leveraging computational complexity theory, APX-hardness analysis, and parameterized complexity techniques, the work establishes for the first time that this problem is not only NP-hard but also APX-hard, even in the scalar parameter case. Nevertheless, the problem becomes polynomial-time solvable when either the number of constraints or the number of decision variables is fixed. These theoretical findings demonstrate that the intrinsic difficulty stems from the number of constraints and decision variables rather than the dimensionality of the parameters. Numerical experiments corroborate the validity of these conclusions.

Abstract

Computing the Lipschitz constant of the solution map of a multi-parametric quadratic program is important for the analysis of optimization-based control. This problem is governed by three factors: the parameter dimension, the number of decision variables, and the number of constraints. While empirical evidence has long suggested exponential complexity, a rigorous complexity-theoretic proof has been lacking. In this paper, we fill this gap by proving that this problem is not only NP-hard but also APX-hard. Furthermore, we reveal that: (a) the problem becomes polynomial-time solvable when the number of constraints or decision variables is fixed; and (b) both NP-hardness and APX-hardness persist even in the scalar parameter case. These results confirm that the complexity stems from the number of constraints and variables, rather than the parameter dimension. Numerical experiments further validate these theoretical findings.

Problem

• Lipschitz constant
• multi-parametric quadratic programs
• APX-hardness
• NP-hardness
• computational complexity

Innovation

• APX-hardness
• multi-parametric quadratic programming
• Lipschitz constant
• computational complexity
• optimization-based control