Probing the Pareto front of a large-scale multiobjective problem with a MIP solver

Abstract

The rapid growth of computing power and the development of highly effective optimization solvers build the appetite for solving increasingly extensive problems. However, despite all these efforts, resource constraints (time, memory) often strike back. The ”curse of dimensionality” haunts primarily combinatorial problems, but not only. The issue is even more acute in multiobjective optimization, where several Pareto optimal solutions have to be derived. In our earlier works, we developed a general methodology for multiobjective optimization that allows representing the outcome of a Pareto optimal solution by a hyperrectangle. The sides of the hyperrectangle are defined by lower and upper bounds on the outcome components, i.e., intervals of possible objective function values. Such a representation makes sense if the Pareto optimal solution cannot be derived with the available computation resources. Beyond the research interest, to be of practical value, methodologies of that kind have to be computationally effective and scalable. In this work, we show that our methodology can be effectively coupled with any MIP optimization solver. With that, as long as an analyst is willing to accept a (sufficiently tight) interval representation of the Pareto optimal solution outcome instead of its exact outcome, our methodology scales multiobjective-based analyses well beyond the reach of the MIP solver itself. We operationalize our methodology in the form of a workflow (we nicknamed it Crescent Workflow). We illustrate the workflow working on several large-scale instances of the multiobjective multidimensional 0–1 knapsack problem with three objectives.

Appendices

Appendix A

See Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22.

There are twenty tables (two tables per each iteration) being the result of Crescent Workflow run as described in Sect. 4.3. In the tables below, to observe tightening of bounds and improvements in $G_{P_{sub},l}\,$, values which have improved in the current iteration are set in italic.

Table 3 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 1, step 1

Table 4 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 1, step 2

Table 5 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 2, step 1

Table 6 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 2, step 2

Table 7 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 3, step 1

Table 8 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 3, step 2

Table 9 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 4, step 1

Table 10 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 4, step 2

Table 11 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 5, step 1

Table 12 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 5, step 2

Table 13 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 6, step 1

Table 14 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 6, step 2

Table 15 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 7, step 1

Table 16 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 7, step 2

Table 17 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 8, step 1

Table 18 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 8, step 2

Table 19 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 9, step 1

Table 20 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 9, step 2

Table 21 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 10, step 1

Table 22 Lower bounds, upper bounds and Pareto suboptimality gaps: iteration 10, step 2

Appendix B

See Tables 23, 24, 25, 26, 27, 28, 29 and 30.

There are remaining eight tables being the result of Crescent Workflow runs for problems 6.11, 6.21, 8.1, 8.11, 8.21, 9.1, 9.11, and 9.21 as described in Sect. 4.4.

Table 23 Problem 6.11—Pareto suboptimality gaps for varying number of vectors $\lambda$

Table 24 Problem 6.21—Pareto suboptimality gaps for varying number of vectors $\lambda$

Table 25 Problem 8.1—Pareto suboptimality gaps for varying number of vectors $\lambda$

Table 26 Problem 8.11—Pareto suboptimality gaps for varying number of vectors $\lambda$

Table 27 Problem 8.21—Pareto suboptimality gaps for varying number of vectors $\lambda$

Table 28 Problem 9.1—Pareto suboptimality gaps for varying number of vectors $\lambda$

Table 29 Problem 9.11—Pareto suboptimality gaps for varying number of vectors $\lambda$

Table 30 Problem 9.21—Pareto suboptimality gaps for varying number of vectors $\lambda$