Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

Abstract

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.

Keywords:

• Linear programming
• interior-point algorithms
• selective addition of inequalities

Acknowledgements

The authors thank the editor-in-chief, the associate editor and several anonymous referees for their careful review of initial versions of this manuscript. The first author also thanks the Rowe School of Business at Dalhousie University for its hospitality during the sabbatical in which this final paper was revised, and the École Polytechnique de Montréal and the GERAD for their support during the summer in which much of its original research was completed.

Notes

No potential conflict of interest was reported by the authors.