A Dual Method for Minimax Quadratic Programming

TL;DR

The paper studies a minimax quadratic programming problem with coupled inequality constraints, formulated as $\min_{x}\max_{y} f(z)$ where $f(z)=\frac{1}{2}z^{T}Gz + c^{T}z$ and $s(z) \le 0$. It develops a dual active-set algorithm that extends the Goldfarb–Idnani framework to the minimax setting by solving a sequence of equality-constrained subproblems, using S-pairs to track active constraints. Under a mild assumption (e.g., a negative semi-definite coupling condition), the method guarantees finite termination via monotonic descent of the objective, with numerically stable implementation based on $Cholesky$ factorization and $Givens$ rotations. Numerical experiments on randomly generated minimax QPs and an adversarial portfolio model demonstrate accuracy, stability, and computational efficiency, validating the proposed approach for practical minimax optimization problems.

Abstract

This paper investigates minimax quadratic programming problems with coupled inequality constraints. By leveraging a duality theorem, we develop a dual algorithm that extends the dual active set method to the minimax setting, transforming the original inequality constrained problem into a sequence of equality constrained subproblems. Under a suitable assumption, we prove that the associated S-pairs do not repeat and that the algorithm terminates in a finite number of iterations, guaranteed by the monotonic decrease of the objective function value. To ensure numerical stability and efficiency, the algorithm is implemented using Cholesky factorization and Givens rotations. Numerical experiments on both randomly generated minimax quadratic programs and illustrative applications demonstrate the accuracy, stability, and computational effectiveness of the proposed algorithm.