Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear programming sensitivity measured by the optimal value worst-case analysis

Milan Hladík

TL;DR

This work addresses how the optimal value of a linear program responds to data perturbations by introducing a worst-case derivative defined through interval LP data. It develops a closed-form expression for this derivative in the nondegenerate case using primal and dual solutions, and provides upper bounds and structural results for degenerate problems, showing that computing the derivative is generally NP-hard but tractable in special cases. A normalized version, the directional derivative d_r, is proposed to achieve scale-invariant sensitivity measures, along with several exact and special-case formulas. Numerical experiments on synthetic and Netlib data illustrate that the derivatives capture LP sensitivity, with large values surfacing for ill-conditioned problems and degenerate cases posing open computational questions and challenges for exact determination.

Abstract

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.

Linear programming sensitivity measured by the optimal value worst-case analysis

TL;DR

This work addresses how the optimal value of a linear program responds to data perturbations by introducing a worst-case derivative defined through interval LP data. It develops a closed-form expression for this derivative in the nondegenerate case using primal and dual solutions, and provides upper bounds and structural results for degenerate problems, showing that computing the derivative is generally NP-hard but tractable in special cases. A normalized version, the directional derivative d_r, is proposed to achieve scale-invariant sensitivity measures, along with several exact and special-case formulas. Numerical experiments on synthetic and Netlib data illustrate that the derivatives capture LP sensitivity, with large values surfacing for ill-conditioned problems and degenerate cases posing open computational questions and challenges for exact determination.

Abstract

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.
Paper Structure (15 sections, 10 theorems, 49 equations, 1 figure, 2 tables)

This paper contains 15 sections, 10 theorems, 49 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Interval data.
  3. Interval linear programming.
  4. Structure of the paper.
  5. Worst-case optimal value derivatives
  6. Worst case type derivative.
  7. The formula for $d_w$.
  8. Degenerate case
  9. Computational complexity
  10. Another approach
  11. A normalized version
  12. Special cases.
  13. Other LP forms
  14. Examples
  15. Conclusion

Key Result

Lemma 1

If lpGen has a unique nondegenerate optimal solution $x^*$ corresponding to a basis $B$, and if $y^*$ is a dual optimal solution, then for a sufficiently small $\alpha>0$ and for every $A'\in\hbox{${\textrm{\boldmath $A$}}$}_\alpha$, $b'\in{\textrm{\boldmath $b$}}_\alpha$ and $c'\in{\textrm{\boldmat

Figures (1)

  • Figure 1: (Example \ref{['exNestab']}) The dual LP problem with the inequality $y_1+y_2\leq c_3$ having the values of $c_3\in\{1.5,\,2,\,2.5\}$.

Theorems & Definitions (20)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • proof
  • Lemma 3
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 10 more