Continuity in Parametric Linear Programming

TL;DR

This paper investigates the continuity and sensitivity properties of linear programming solutions and optimal values when all parameters vary, providing theoretical insights into their stability and behavior.

Contribution

It establishes new results on the upper and lower semicontinuity of feasible and solution correspondences, and on the continuity of optimal value functions in parametric linear programming.

Findings

Feasible and solution correspondences are upper and lower semicontinuous under parameter variations.

Optimal value functions exhibit continuity properties when all parameters vary.

Sensitivity analysis is performed with fixed coefficient matrices, revealing stability conditions.

Abstract

In this paper we assemble some results about the upper-semicontinuity and lower-semicontinuity of the feasible correspondence and the solution correspondence of linear programming problems allowing variability of all parameters of such problems. We also prove continuity properties of optimal value functions, once again allowing all parameters to vary. We discuss sensitivity properties of the optimal value function, keeping the coefficient matrix fixed.