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A Tunneling Method for Nonlinear Multi-objective Optimization Problems

Bikram Adhikary, Md Abu Talhamainuddin Ansary

TL;DR

This work addresses nonlinear multi-objective optimization with box constraints by introducing a parameter-free multi-objective tunneling method that alternates a minimization phase with a tunneling phase guided by a multiobjective auxiliary function. A reference point $z^*$ is used to construct a family of tunneling functions $\mathbb{T}_k$, producing new critical points $\bar{z}$ with improved objective values and enabling iterative enrichment of the global Pareto front. A central theoretical result proves that any critical point of the tunneling problem is also a critical point of the original problem $MOP_{BC}$, ensuring validity of successive tunneling steps. Empirical results on 42 benchmarks show that MOTM yields broader Pareto front coverage and higher hypervolume compared to MOSQCQP and NSGA-II, confirming robustness and practical value with an eye toward extending to constrained and nonsmooth problems.

Abstract

In this paper, a tunneling method is developed for nonlinear multiobjective optimization problems using some ideas of the single objective tunneling method. The proposed method does not require any a priori chosen parameters or ordering information of the objective functions. At any critical point, an auxiliary function is developed to find a different critical point that dominates the previous one. By repeatedly applying the tunneling procedure, it is possible to construct a broader approximation to the global Pareto front in nonconvex multi-objective optimization problems that may contain multiple local Pareto fronts. An algorithm is then designed based on this auxiliary function, and the convergence of this algorithm is justified under some mild assumptions. Finally, several numerical examples are presented to illustrate the effectiveness of the proposed method and to justify the theoretical results.

A Tunneling Method for Nonlinear Multi-objective Optimization Problems

TL;DR

This work addresses nonlinear multi-objective optimization with box constraints by introducing a parameter-free multi-objective tunneling method that alternates a minimization phase with a tunneling phase guided by a multiobjective auxiliary function. A reference point is used to construct a family of tunneling functions , producing new critical points with improved objective values and enabling iterative enrichment of the global Pareto front. A central theoretical result proves that any critical point of the tunneling problem is also a critical point of the original problem , ensuring validity of successive tunneling steps. Empirical results on 42 benchmarks show that MOTM yields broader Pareto front coverage and higher hypervolume compared to MOSQCQP and NSGA-II, confirming robustness and practical value with an eye toward extending to constrained and nonsmooth problems.

Abstract

In this paper, a tunneling method is developed for nonlinear multiobjective optimization problems using some ideas of the single objective tunneling method. The proposed method does not require any a priori chosen parameters or ordering information of the objective functions. At any critical point, an auxiliary function is developed to find a different critical point that dominates the previous one. By repeatedly applying the tunneling procedure, it is possible to construct a broader approximation to the global Pareto front in nonconvex multi-objective optimization problems that may contain multiple local Pareto fronts. An algorithm is then designed based on this auxiliary function, and the convergence of this algorithm is justified under some mild assumptions. Finally, several numerical examples are presented to illustrate the effectiveness of the proposed method and to justify the theoretical results.
Paper Structure (6 sections, 2 theorems, 14 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 2 theorems, 14 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multi-objective tunneling method
  4. Algorithm
  5. Numerical illustration and discussion
  6. Conclusion

Key Result

Theorem 2.1

Let $z^* \in \mathcal{Z}$ be a weak efficient solution of $(\mathrm{MOP}_{BC})$. Then there exist multipliers $(\lambda, \mu) \in \mathbb{R}_{+}^{m} \times \mathbb{R}_{+}^{2n}$, $(\lambda, \mu) \neq \mathbf{0}^{m+2n}$, such that where $e_1=(1,0,0,\dots,0)^T$.

Figures (5)

  • Figure 1: Performance profiles using purity metric
  • Figure 2: Performance profiles using $\Gamma$-spread metric
  • Figure 3: Performance profiles using $\Delta$-spread metric
  • Figure 4: Performance profiles using hypervolume metric
  • Figure 5: Performance profiles using number of function evaluations

Theorems & Definitions (5)

  • Theorem 2.1: First-order necessary condition for weak efficiency
  • Definition 1: Critical point
  • Theorem 3.1
  • proof
  • Remark 1