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Range of optimal values in absolute value linear programming with interval data

Milan Hladík

TL;DR

This work studies absolute value linear programming with interval data, focusing on the range of optimal values under data uncertainty. It derives a closed form for the best-case optimum by reducing to a single real AVLP via orthant decomposition, and develops lower and upper bound strategies for the worst-case optimum, including an iterative refinement procedure. The analysis also introduces basis stability concepts to simplify the best-case computation and relates the worst-case AVLP to generalized absolute value equations. The results provide insight into robustness of AVLP under data intervals and highlight open questions on complexity and solution-set descriptions. Practical implications include tractable methods for estimating the best and worst outcomes and guidance on stability regimes in interval optimization.

Abstract

Absolute value linear programming problems is quite a new area of optimization problems, involving linear functions and absolute values in the description of the model. In this paper, we consider interval uncertainty of the input coefficients. Our goal is to determine the best and the worst case optimal values. For the former, we derive an explicit formula, reducing the problem to a certain optimization problem. However, the latter is more complicated, and we propose a lower and upper bound approaches to estimate the value. We also investigate the basis stability, in which situation the best case optimal value is efficiently computable. The worst case optimal value then also admits a simple characterization; however, the computational complexity remains open.

Range of optimal values in absolute value linear programming with interval data

TL;DR

This work studies absolute value linear programming with interval data, focusing on the range of optimal values under data uncertainty. It derives a closed form for the best-case optimum by reducing to a single real AVLP via orthant decomposition, and develops lower and upper bound strategies for the worst-case optimum, including an iterative refinement procedure. The analysis also introduces basis stability concepts to simplify the best-case computation and relates the worst-case AVLP to generalized absolute value equations. The results provide insight into robustness of AVLP under data intervals and highlight open questions on complexity and solution-set descriptions. Practical implications include tractable methods for estimating the best and worst outcomes and guidance on stability regimes in interval optimization.

Abstract

Absolute value linear programming problems is quite a new area of optimization problems, involving linear functions and absolute values in the description of the model. In this paper, we consider interval uncertainty of the input coefficients. Our goal is to determine the best and the worst case optimal values. For the former, we derive an explicit formula, reducing the problem to a certain optimization problem. However, the latter is more complicated, and we propose a lower and upper bound approaches to estimate the value. We also investigate the basis stability, in which situation the best case optimal value is efficiently computable. The worst case optimal value then also admits a simple characterization; however, the computational complexity remains open.

Paper Structure

This paper contains 19 sections, 10 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Absolute value linear programming.
  3. Notation.
  4. Interval computation.
  5. Preliminaries.
  6. Interval AVLP problem.
  7. The goal: Range of optimal values.
  8. General results
  9. Best case optimal value.
  10. Worst case optimal value.
  11. Worst case optimal value: lower bound.
  12. Worst case optimal value: upper bound.
  13. Worst case optimal value: examples.
  14. Basis stability
  15. Basis in LP.
  16. ...and 4 more sections

Key Result

Proposition 1

We have

Figures (4)

  • Figure 1: (Example \ref{['exUfUflDiff']}) Illustration of the feasible set with different choices of $a\in\bm{a}$
  • Figure 2: (Example \ref{['exUfFinAttain']}) Illustration of the feasible set with different choices of $a\in\bm{a}$
  • Figure 3: (Example \ref{['exUfInfInfeas']}) Illustration of the feasible set with different choices of $a\in\bm{a}$
  • Figure 4: (Example \ref{['exBstabUf']}) The grey area is the feasible set of the nominal values; the rectangle is an enclosure $\bm{x}$ of the optimal values; and the diamond-like shape is the real optimal solution set

Theorems & Definitions (24)

  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Example 1
  • Example 2
  • ...and 14 more