Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A general framework for providing interval representations of Pareto optimal outcomes for large-scale bi- and tri-criteria MIP problems

Grzegorz Filcek, Janusz Miroforidis

TL;DR

This work proposes an algorithm to provide the so-called interval representation of the Pareto optimal outcome designated by the weighting vector when there is a time limit on solving the Chebyshev scalarization of the Multi-Objective Mixed-Integer Programming problem.

Abstract

The Multi-Objective Mixed-Integer Programming (MOMIP) problem is one of the most challenging. To derive its Pareto optimal solutions one can use the well-known Chebyshev scalarization and Mixed-Integer Programming (MIP) solvers. However, for a large-scale instance of the MOMIP problem, its scalarization may not be solved to optimality, even by state-of-the-art optimization packages, within the time limit imposed on the optimization. If a MIP solver cannot derive the optimal solution within the assumed time limit, it provides the optimality gap, which gauges the quality of the approximate solution. However, for the MOMIP case, no information is provided on the lower and upper bounds of the components of the Pareto optimal outcome. For the MOMIP problem with two and three objective functions, an algorithm is proposed to provide the so-called interval representation of the Pareto optimal outcome designated by the weighting vector when there is a time limit on solving the Chebyshev scalarization. Such interval representations can be used to navigate on the Pareto front. The results of several numerical experiments on selected large-scale instances of the multi-objective, multidimensional 0-1 knapsack problem illustrate the proposed approach. The limitations and possible enhancements of the proposed method are also discussed.

A general framework for providing interval representations of Pareto optimal outcomes for large-scale bi- and tri-criteria MIP problems

TL;DR

This work proposes an algorithm to provide the so-called interval representation of the Pareto optimal outcome designated by the weighting vector when there is a time limit on solving the Chebyshev scalarization of the Multi-Objective Mixed-Integer Programming problem.

Abstract

The Multi-Objective Mixed-Integer Programming (MOMIP) problem is one of the most challenging. To derive its Pareto optimal solutions one can use the well-known Chebyshev scalarization and Mixed-Integer Programming (MIP) solvers. However, for a large-scale instance of the MOMIP problem, its scalarization may not be solved to optimality, even by state-of-the-art optimization packages, within the time limit imposed on the optimization. If a MIP solver cannot derive the optimal solution within the assumed time limit, it provides the optimality gap, which gauges the quality of the approximate solution. However, for the MOMIP case, no information is provided on the lower and upper bounds of the components of the Pareto optimal outcome. For the MOMIP problem with two and three objective functions, an algorithm is proposed to provide the so-called interval representation of the Pareto optimal outcome designated by the weighting vector when there is a time limit on solving the Chebyshev scalarization. Such interval representations can be used to navigate on the Pareto front. The results of several numerical experiments on selected large-scale instances of the multi-objective, multidimensional 0-1 knapsack problem illustrate the proposed approach. The limitations and possible enhancements of the proposed method are also discussed.
Paper Structure (17 sections, 3 theorems, 7 equations, 4 figures, 28 tables, 2 algorithms)

This paper contains 17 sections, 3 theorems, 7 equations, 4 figures, 28 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Lower and upper bounds on components of implicit Pareto optimal outcomes
  4. The interval representation of the implicit Pareto optimal outcome
  5. Providing interval representations of implicit Pareto optimal outcomes
  6. The derivation of lower and upper shells
  7. Calculating interval representations
  8. Computational experiments
  9. Multi-objective multidimensional 0-1 knapsack problem
  10. Test instances of the MOMIP problem
  11. Experimental setting
  12. Experiment 1
  13. Experiment 2
  14. Comparing Chute2 with Chute1
  15. Discussion
  16. ...and 2 more sections

Key Result

Lemma 1

Given lower shell $S_L$ and upper shell $S_U$. Suppose $x \in S_U$ and $L_{\,\bar{l}\,}(S_L, \lambda) \leqslant f_{\,\bar{l}\,}(x)$ for some $\bar{l}$ and $L_l(S_L, \lambda) \geqslant f_l(x)$ for all $l=1,\ldots,k, \ l \neq \bar{l}$. Then $x$ provides an upper bound for $f_{\,\bar{l}\,}(x^{P_{opt}}

Figures (4)

  • Figure 1: Components of $R(S_L, S_U,\lambda)$: $\square$, $\circ$ -- images of lower shell $S_L$ and upper shell $S_U$ elements, respectively, in the objective space, $\blacktriangle$ -- vector of lower bounds, $\blacksquare$ -- vector of upper bounds.
  • Figure 2: Deriving upper shell $S^1_U$ whose element $x^{{\lambda}^{"'}}$ is a source of an upper bound, $U_1$, for $f_{1}(x^{P_{opt}}(\lambda))$ with some $\lambda$ : $\circ$ -- image of upper shell $S^1_U$ in the objective space, $\blacktriangle$ -- vector of lower bounds.
  • Figure 3: The idea of deriving upper shell $\{x(\tilde{\mu})\}$ whose element is a source of a better upper bound on $f_{2}(x^{P_{opt}}(\lambda))$ than the element of upper shell $\{x(\mu^{I})\}$.
  • Figure 4: A finite two-sided approximation of the Pareto front: $\square$, $\circ$ -- images of lower shell $S_L$ and upper shell $S_U$ elements in the objective space, respectively, $\bullet$ -- $y^{*}$.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3