Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming

• Published in: Journal of optimization theory and applications Vol. 181; no. 3; pp. 905 - 945
• Main Authors: Pougkakiotis, Spyridon, Gondzio, Jacek
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Eigenvalues; Engineering; Experimentation; Iterative methods; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Operations Research/Decision Theory; Optimization; Perturbation; Properties (attributes); Quadratic programming; Regularization; Theory of Computation; Interior point methods; Generalized proximal point methods; Primal–dual regularization
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-019-01491-1

In this paper, we present a dynamic non-diagonal regularization for interior point methods. The non-diagonal aspect of this regularization is implicit, since all the off-diagonal elements of the regularization matrices are cancelled out by those elements present in the Newton system, which do not contribute important information in the computation of the Newton direction. Such a regularization has multiple goals. The obvious one is to improve the spectral properties of the Newton system solved at each iteration of the interior point method. On the other hand, the regularization matrices introduce sparsity to the aforementioned linear system, allowing for more efficient factorizations. We also propose a rule for tuning the regularization dynamically based on the properties of the problem, such that sufficiently large eigenvalues of the non-regularized system are perturbed insignificantly. This alleviates the need of finding specific regularization values through experimentation, which is the most common approach in the literature. We provide perturbation bounds for the eigenvalues of the non-regularized system matrix and then discuss the spectral properties of the regularized matrix. Finally, we demonstrate the efficiency of the method applied to solve standard small- and medium-scale linear and convex quadratic programming test problems.