Constraint augmentation in pseudo-singularly perturbed linear programs

• Published in: Mathematical programming Vol. 132; no. 1-2; pp. 179 - 208
• Main Authors: Avrachenkov, K., Burachik, R. S., Filar, J. A., Gaitsgory, V.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.04.2012; Springer; Springer Nature B.V
• Subjects: Analysis; Applied mathematics; Applied sciences; Asymptotic methods; Asymptotic properties; Augmentation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Constraining; Exact sciences and technology; Full Length Paper; Iterative methods; Linear programming; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Perturbation methods; Studies; Theoretical; 90C31; 90C05; Discontinuity; Sensitivity analysis; Constraint; Singular perturbation; Iterative method; Linear programming; Modeling; Constrained optimization; Lagrange multiplier; Perturbation method; Asymptotic approximation; Objective function
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-010-0388-0

In this paper we study a linear programming problem with a linear perturbation introduced through a parameter ε > 0. We identify and analyze an unusual asymptotic phenomenon in such a linear program. Namely, discontinuous limiting behavior of the optimal objective function value of such a linear program may occur even when the rank of the coefficient matrix of the constraints is unchanged by the perturbation. We show that, under mild conditions, this phenomenon is a result of the classical Slater constraint qualification being violated at the limit and propose an iterative, constraint augmentation approach for resolving this problem.