Towards an efficient augmented Lagrangian method for convex quadratic programming

• Published in: Computational optimization and applications Vol. 76; no. 3; pp. 767 - 800
• Main Authors: Bueno, Luís Felipe, Haeser, Gabriel, Santos, Luiz-Rafael
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2020; Springer Nature B.V
• Subjects: Convex and Discrete Geometry; Lagrange multiplier; Management Science; Mathematics; Mathematics and Statistics; Methods; Nonlinear programming; Operations Research; Operations Research/Decision Theory; Optimization; Quadratic equations; Quadratic programming; Queuing theory; Sensitivity analysis; Statistics; 49K99; 90C51; 65K05; 90C30; Augmented Lagrangian; Convex quadratic programming; Linear programming; Interior point methods; 90C05
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-019-00161-2

Interior point methods have attracted most of the attention in the recent decades for solving large scale convex quadratic programming problems. In this paper we take a different route as we present an augmented Lagrangian method for convex quadratic programming based on recent developments for nonlinear programming. In our approach, box constraints are penalized while equality constraints are kept within the subproblems. The motivation for this approach is that Newton’s method can be efficient for minimizing a piecewise quadratic function. Moreover, since augmented Lagrangian methods do not rely on proximity to the central path, some of the inherent difficulties in interior point methods can be avoided. In addition, a good starting point can be easily exploited, which can be relevant for solving subproblems arising from sequential quadratic programming, in sensitivity analysis and in branch and bound techniques. We prove well-definedness and finite convergence of the method proposed. Numerical experiments on separable strictly convex quadratic problems formulated from the Netlib collection show that our method can be competitive with interior point methods, in particular when a good initial point is available and a second-order Lagrange multiplier update is used.