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

Abstract

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.

A new preconditioning approach for an interior point‐proximal method of multipliers for linear and convex quadratic programming

“…In particular, we solve each set with increasing accuracy and report the overall success rate of the method, the total time, as well as the total IP-PMM and Krylov iterations. All previous experiments demonstrate that IP-PMM with the proposed preconditioning strategy inherits the reliability of IP-PMM with a direct approach (factorization), 8 while allowing one to control the memory and processing requirements of the method (which is not the case when employing a factorization to solve the resulting Newton systems). Most of the previous experiments were conducted on small-to medium-scale linear and convex quadratic programming problems.…”

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

“…In this paper, we are extending the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in [19]. In particular, the algorithm in [19] was developed for convex quadratic programming problems and assumed that the resulting linear systems are solved exactly.…”

“…In this paper, we are extending the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in [19]. In particular, the algorithm in [19] was developed for convex quadratic programming problems and assumed that the resulting linear systems are solved exactly. Under this framework, it was proved that IP-PMM converges in a polynomial number of iterations, under mild assumptions, and an infeasibility detection mechanism was established.…”

“…Under this framework, it was proved that IP-PMM converges in a polynomial number of iterations, under mild assumptions, and an infeasibility detection mechanism was established. An important feature of this method is that it provides a reliable tuning for the penalty parameters of the PMM; indeed, the reliability of the algorithm is established numerically in a wide variety of convex problems in [7,8,10,19]. In particular, the IP-PMMs proposed in [7,8,10] use preconditioned iterative methods for the solution of the resulting linear systems, and are very robust despite the use of inexact Newton directions.…”

“…We show that the method converges polynomially under standard assumptions. Subsequently, we provide a necessary condition for lack of strong duality, which can serve as a basis for constructing implementable detection mechanisms for pathological cases (following the developments in [19]). As is verified in [19], IP-PMM is competitive with standard non-regularized IPM schemes, and is significantly more robust.…”