An exact primal–dual penalty method approach to warmstarting interior-point methods for linear programming

Abstract

Abstract. One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primal-dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems.

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

“…To solve the NLP, an interior point method is used, involving the warmstart scheme of Benson and Shanno [50][51].…”

Using Interior-Point Methods within an Outer Approximation Framework for Mixed Integer Nonlinear Programming

“…We resolved the issues of warmstarting, infeasibility detection, and robustness for the interior-point method. In doing so, we used the exact primal-dual penalty method of [7] and [8]. The resulting algorithm was implemented using the interior-point code MILANO [4] and tested on a suite of MINLPs.…”

“…Numerical results in [7] and [8] demonstrate the strong performance of the primal-dual penalty approach under a variety of problem modifications, including the addition of constraints and variables. Thus, we are optimistic that the performance improvements demonstrated in this paper will continue to be applicable when used within any integer programming framework.…”

“…However, they have other desirable properties, including infeasibility detection capabilities and regularizations that automatically satisfy constraint qualifications. The exact primal-dual penalty method proposed in [7] for linear programming and in [8] for nonlinear programming is a remedy that has been demonstrated to work for warmstarts. This method relaxes the nonnegativity constraint on the dual and slack variables and provides regularization for the matrix in the reduced KKT system (3.7).…”

“…These relaxation variables are incorporated into the objective function using a penalty term with dual penalty parameters u. For further details of the primal and dual problems, as well as a proof of the exactness of the penalty approach, the reader is referred to [7] and [8].…”