An Efficient Solution Method for Solving Convex Separable Quadratic Optimization Problems

Shaoze Li, Junhao Wu, Cheng Lu, Zhibin Deng, Shu-Cherng Fang

TL;DR

This work tackles convex separable quadratic optimization with multiple separable quadratic constraints, formulated as min_{m{y}} f(m{y}) = m{y}^T \Delta \bm{y} + \bm{\alpha}^T \bm{y} subject to g_i(\bm{y}) = \bm{y}^T \Theta_i \bm{y} + \bm{\beta}_i^T \bm{y} + \sigma_i \le 0 and \bm{y} \in [\bm{l}, \bm{u}], where \Delta and \Theta_i are diagonal. The authors derive an iterative resolution of the KKT system that yields a dual coordinate ascent algorithm with a convergence proof and demonstrate its superior performance over Gurobi on large-scale problems. The core contribution is a scalable solver that reduces to a closed-form inner update for the single-constraint case and extends to multiple constraints via cyclic dual updates, with rigorous convergence analysis. Practically, the method delivers fast solutions for large n with a modest number of quadratic constraints, highlighting its applicability to large-scale resource allocation and network-flow problems.

Abstract

Convex separable quadratic optimization problems occur in many practical applications. In this paper, based on an iterative resolution scheme of the KKT system, we develop an efficient method for solving a quadratic programming problem with a convex separable objective function subject to multiple convex separable constraints. We show that the proposed approach leads to a dual coordinate ascent algorithm and provide a convergence proof. Numerical experiments support the superior performance of the proposed method to that of the Gurobi solver, especially for solving large-scale convex separate quadratic programming problems.

An Efficient Solution Method for Solving Convex Separable Quadratic Optimization Problems

TL;DR

This work tackles convex separable quadratic optimization with multiple separable quadratic constraints, formulated as min_{m{y}} f(m{y}) = m{y}^T \Delta \bm{y} + \bm{\alpha}^T \bm{y} subject to g_i(\bm{y}) = \bm{y}^T \Theta_i \bm{y} + \bm{\beta}_i^T \bm{y} + \sigma_i \le 0 and \bm{y} \in [\bm{l}, \bm{u}], where \Delta and \Theta_i are diagonal. The authors derive an iterative resolution of the KKT system that yields a dual coordinate ascent algorithm with a convergence proof and demonstrate its superior performance over Gurobi on large-scale problems. The core contribution is a scalable solver that reduces to a closed-form inner update for the single-constraint case and extends to multiple constraints via cyclic dual updates, with rigorous convergence analysis. Practically, the method delivers fast solutions for large n with a modest number of quadratic constraints, highlighting its applicability to large-scale resource allocation and network-flow problems.

Abstract

Convex separable quadratic optimization problems occur in many practical applications. In this paper, based on an iterative resolution scheme of the KKT system, we develop an efficient method for solving a quadratic programming problem with a convex separable objective function subject to multiple convex separable constraints. We show that the proposed approach leads to a dual coordinate ascent algorithm and provide a convergence proof. Numerical experiments support the superior performance of the proposed method to that of the Gurobi solver, especially for solving large-scale convex separate quadratic programming problems.