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

Abstract

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 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.

An interior point-proximal method of multipliers for convex quadratic programming

“…There, global convergence of the method was proved, under some assumptions. A variation of the method proposed in [12] is given in [29], where general non-diagonal regularization matrices are employed, as a means of further improving factorization efficiency.…”

“…This value is based on the developments in [29], in order to ensure that we introduce a controlled perturbation in the eigenvalues of the non-regularized linear system. The reader is referred to [29] for an extensive study on the subject. If the factorization fails, we increase the regularization parameters by a factor of 10 and repeat the factorization.…”

Regularised primal–dual interior‐point method for dynamic optimal power flow with block‐angular structures

“…This is because of the disturbance factor μ k + 1 → 0, and we force the complementarity conditions to be approximately satisfied L r V r e ≃ μe and U r W r e ≃ − μe. Therefore, the matrix ℳ r becomes extremely illconditioned [49].…”

“…The value of parameter ρ is tuned experimentally over a variety of problems. In general, the goals of regularisation matrices for PDIPM are [48][49][50][51][52][53][54][55][56][57]:…”

An Interior Point-Proximal Method of Multipliers for Linear Positive Semi-Definite Programming

“…This is because the introduction of regularization prevents severe ill-conditioning and rank deficiency of the associated linear systems solved within standard IPMs, which can hinder their convergence and numerical stability. For detailed discussions on the effectiveness of regularization within IPMs, the reader is referred to [1,2,18], and the references therein. A particularly important benefit of using regularization, is that the resulting Newton systems can be preconditioned effectively (e.g.…”