2023/07/13 by Milan Hladík, Hladík, Milan
Decision Sciences · Engineering · Mathematics · #Advanced Control Systems Optimization #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Risk and Portfolio Optimization
paper · pdf · doi:10.48550/arxiv.2307.06733
openalex publication_date 2023/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper introduces a concept of a derivative of the optimal value function in linear programming (LP). Basically, it is the the worst case optimal value of an interval LP problem when the nominal data the data are inflated to intervals according to given perturbation patterns. By definition, the derivative expresses how the optimal value can worsen when the data are subject to variation. In addition, it also gives a certain sensitivity measure or condition number of an LP problem. If the LP problem is nondegenerate, the derivatives are easy to calculate from the computed primal and dual optimal solutions. For degenerate problems, the computation is more difficult. We propose an upper bound and some kind of characterization, but there are many open problems remaining. We carried out numerical experiments with specific LP problems and with real LP data from Netlib repository. They show that the derivatives give a suitable sensitivity measure of LP problems. It remains an open problem how to efficiently and rigorously handle degenerate problems.