Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Trust Region Proximal Gradient Method for Nonlinear Multi-objective Optimization Problems

Md Abu Talhamainuddin Ansary

Abstract

In this paper, a globally convergent trust region proximal gradient method is developed for composite multi-objective optimization problems where each objective function can be represented as the sum of a smooth function and a nonsmooth function. The proposed method is free from any kind of priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a sub problem is solved to find a suitable direction. This sub problem uses a quadratic approximation of each smooth function and a trust region constraint. An update formula for trust region radius is introduce in this paper. A sequence is generated using descent directions. It is justified that under some mild assumptions every accumulation point of this sequence is a critical point. The proposed method is verified and compared with some existing methods using a set of problems.

A Trust Region Proximal Gradient Method for Nonlinear Multi-objective Optimization Problems

Abstract

In this paper, a globally convergent trust region proximal gradient method is developed for composite multi-objective optimization problems where each objective function can be represented as the sum of a smooth function and a nonsmooth function. The proposed method is free from any kind of priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a sub problem is solved to find a suitable direction. This sub problem uses a quadratic approximation of each smooth function and a trust region constraint. An update formula for trust region radius is introduce in this paper. A sequence is generated using descent directions. It is justified that under some mild assumptions every accumulation point of this sequence is a critical point. The proposed method is verified and compared with some existing methods using a set of problems.

Paper Structure

This paper contains 7 sections, 8 theorems, 61 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A trust region proximal gradient method for $(MOP)$
  4. Algorithm
  5. Convergence analysis of Algorithm \ref{['trpxm1']}
  6. Numerical examples
  7. Conclusion

Key Result

Theorem 1

( Theorem 3.14, proxg1) Let $h:\mathbb{R}^n\rightarrow (-\infty,\infty]$ be a proper convex function, and assume that $x\in int(dom~h)$. Then $\partial h(x)$ is nonempty and bounded.

Figures (8)

  • Figure 1: Convex hull of $\left\{\partial F_1(x^*),\partial F_2(x^*)\right\}$
  • Figure 2: Approximate Pareto fronts of $P_1$
  • Figure 3: Approximate Pareto fronts of (MOLS)
  • Figure 4: Performance profiles using purity metric
  • Figure 5: Performance profiles using $\Gamma$-spread metric
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 3