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

TL;DR

This paper investigates how common transformations in linear programming affect the set of optimal solutions and value ranges in interval linear programming, especially under uncertainty, and identifies cases with stronger guarantees.

Contribution

It analyzes the impact of standard LP transformations on interval programs and introduces results for a special class with uncertainty only in the objective and right-hand side.

Findings

Transformations can alter the set of optimal solutions and value ranges in interval LP.

Certain properties do not hold generally but are valid for a special class with limited uncertainty.

Stronger results are obtained when uncertainty affects only the objective and right-hand side.

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.