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On the convergence of conditional gradient method for unbounded multiobjective optimization problems

Wang Chen, Yong Zhao, Liping Tang, Xinmin Yang

Abstract

This paper focuses on developing a conditional gradient algorithm for multiobjective optimization problems with an unbounded feasible region. We employ the concept of recession cone to establish the well-defined nature of the algorithm. The asymptotic convergence property and the iteration-complexity bound are established under mild assumptions. Numerical examples are provided to verify the algorithmic performance.

On the convergence of conditional gradient method for unbounded multiobjective optimization problems

Abstract

This paper focuses on developing a conditional gradient algorithm for multiobjective optimization problems with an unbounded feasible region. We employ the concept of recession cone to establish the well-defined nature of the algorithm. The asymptotic convergence property and the iteration-complexity bound are established under mild assumptions. Numerical examples are provided to verify the algorithmic performance.
Paper Structure (5 sections, 9 theorems, 36 equations, 2 figures, 1 table)

This paper contains 5 sections, 9 theorems, 36 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The conditional gradient algorithm
  4. Convergence analysis
  5. Numerical examples

Key Result

Lemma 2.1

assunccao2021conditional If $F$ is convex on $\Omega$ and $\bar{x}\in\Omega$ is a Pareto critical point, then $\bar{x}$ is also a weak Pareto optimal solution of mop.

Figures (2)

  • Figure 1: Visualization of the objective functions $F$ on Examples \ref{['ex1']} and \ref{['ex2']}.
  • Figure 2: The final solutions and the paths of iterations obtained by the algorithm on the two examples.

Theorems & Definitions (15)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.1
  • Lemma 4.1
  • Theorem 4.1
  • ...and 5 more