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Interval Linear Programming under Transformations: Optimal Solutions and Optimal Value Range

Elif Garajová, Milan Hladík, Miroslav Rada

TL;DR

This work analyzes how standard linear programming transformations interact with interval linear programs (ILPs) under the dependency problem, where interval coeffients perturb independently. It shows that while some transformations, like changing the objective or adding slack variables, can be applied without altering core properties, others—such as splitting equations or imposing non-negativity—can change the optimal solution set and, in some cases, the optimal value range. A key result is that for ILPs with a fixed coefficient matrix, all considered transformations preserve the optimal solution set, though the optimal value range may still change due to infeasibility or unboundedness. The paper also derives conditions under which the best-case bound remains invariant under certain transformations and discusses how duality informs the computation of optimal value ranges, offering guidance for modeling under uncertainty. Overall, the findings clarify the limits of applying classical LP transformations to ILPs and highlight the dependency problem as a central consideration in interval optimization.

Abstract

Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb independently within the given lower and upper bounds. However, contrarily to classical linear programming, an interval program cannot always be converted into a desired form without affecting its properties, due to the so-called dependency problem. In this paper, we discuss the common transformations used in linear programming, such as imposing non-negativity on free variables or splitting equations into inequalities, and their effects on interval programs. Specifically, we examine changes in the set of all optimal solutions, optimal values and the optimal value range. Since some of the considered properties do not holds in the general case, we also study a special class of interval programs, in which uncertainty only affects the objective function and the right-hand-side vector. For this class, we obtain stronger results.

Interval Linear Programming under Transformations: Optimal Solutions and Optimal Value Range

TL;DR

This work analyzes how standard linear programming transformations interact with interval linear programs (ILPs) under the dependency problem, where interval coeffients perturb independently. It shows that while some transformations, like changing the objective or adding slack variables, can be applied without altering core properties, others—such as splitting equations or imposing non-negativity—can change the optimal solution set and, in some cases, the optimal value range. A key result is that for ILPs with a fixed coefficient matrix, all considered transformations preserve the optimal solution set, though the optimal value range may still change due to infeasibility or unboundedness. The paper also derives conditions under which the best-case bound remains invariant under certain transformations and discusses how duality informs the computation of optimal value ranges, offering guidance for modeling under uncertainty. Overall, the findings clarify the limits of applying classical LP transformations to ILPs and highlight the dependency problem as a central consideration in interval optimization.

Abstract

Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb independently within the given lower and upper bounds. However, contrarily to classical linear programming, an interval program cannot always be converted into a desired form without affecting its properties, due to the so-called dependency problem. In this paper, we discuss the common transformations used in linear programming, such as imposing non-negativity on free variables or splitting equations into inequalities, and their effects on interval programs. Specifically, we examine changes in the set of all optimal solutions, optimal values and the optimal value range. Since some of the considered properties do not holds in the general case, we also study a special class of interval programs, in which uncertainty only affects the objective function and the right-hand-side vector. For this class, we obtain stronger results.

Paper Structure

This paper contains 14 sections, 9 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Interval Linear Programming
  3. Transformations of Interval Linear Programs
  4. Changing the objective.
  5. Adding slack variables.
  6. Splitting equations into inequalities.
  7. Imposing non-negativity.
  8. Optimal Solution Set under Transformations
  9. General Case
  10. Special Case: Fixed Coefficient Matrix
  11. Optimal Values under Transformations
  12. General Case
  13. Special Case: Fixed Coefficient Matrix
  14. Conclusion

Key Result

Theorem 1

A vector $x \in \mathbb{R}^n$ is a weakly feasible solution of an interval linear system $\boldsymbol{A}x = \boldsymbol{b}$ if and only if it is a weakly feasible solution of the system $\boldsymbol{A}x \le \boldsymbol{b}, \boldsymbol{A}x \ge \boldsymbol{b}$.

Figures (2)

  • Figure 1: Transformations preserving the feasible (dashed arrows) and optimal (solid arrows) solution set of a general interval linear program.
  • Figure 2: Transformations preserving the optimal solution set of an interval linear program with a fixed coefficient matrix.

Theorems & Definitions (19)

  • Theorem 1: Li:2015
  • Example 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 1: continued
  • Theorem 4: Hladik:2009
  • Theorem 5
  • proof
  • ...and 9 more