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Generalized conditional gradient methods for multiobjective composite optimization problems with H{ö}lder condition

Wang Chen, Liping Tang, Xinmin Yang

Abstract

In this paper, we deal with multiobjective composite optimization problems, where each objective function is a combination of smooth and possibly non-smooth functions. We first propose a parameter-dependent conditional gradient method to solve this problem. The step size in this method requires prior knowledge of the parameters related to the H{ö}lder continuity of the gradient of the smooth function. The convergence properties of this method are then established. Given that these parameters may be unknown or, if known, may not be unique, the first method may encounter implementation challenges or slow convergence. To address this, we further propose a parameter-free conditional gradient method that determines the step size using a local quadratic upper approximation and an adaptive line search strategy, eliminating the need for any problem-specific parameters. The performance of the proposed methods is demonstrated on several test problems involving the indicator function and an uncertainty function.

Generalized conditional gradient methods for multiobjective composite optimization problems with H{ö}lder condition

Abstract

In this paper, we deal with multiobjective composite optimization problems, where each objective function is a combination of smooth and possibly non-smooth functions. We first propose a parameter-dependent conditional gradient method to solve this problem. The step size in this method requires prior knowledge of the parameters related to the H{ö}lder continuity of the gradient of the smooth function. The convergence properties of this method are then established. Given that these parameters may be unknown or, if known, may not be unique, the first method may encounter implementation challenges or slow convergence. To address this, we further propose a parameter-free conditional gradient method that determines the step size using a local quadratic upper approximation and an adaptive line search strategy, eliminating the need for any problem-specific parameters. The performance of the proposed methods is demonstrated on several test problems involving the indicator function and an uncertainty function.

Paper Structure

This paper contains 8 sections, 15 theorems, 82 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Some notations and basic results
  4. The gap function
  5. A parameter-dependent generalized multiobjective conditional gradient method
  6. A parameter-free multiobjective generalized conditional gradient method
  7. Numerical experiments
  8. Conclusions

Key Result

Lemma 2.1

ito2023parameter Suppose that $\{\beta_{k}\}$ and $\{\gamma_{k}\}$ are sequences of non-negative real numbers such that for all $k\geq0$ and for some constants $c\in(0,1)$, $\alpha>0$ and $A>0$. Assume additionally that $\beta_{k}\geq\gamma_{k}$ for all $k\geq0$ and let $\beta_{k}^{*}=\min_{0\leq\ell\leq k}\beta_{\ell}$. Then for all $k\geq k_{0}$ and for all $k\geq k_{0}+2A/(c\gamma_{k_{0}}^{\

Figures (2)

  • Figure 1: Purity and Spread performance profiles of PGM-CondG and FGM-CondG for the case where $G$ corresponds to Case i.
  • Figure 2: Purity and Spread performance profiles of PGM-CondG and FGM-CondG for the case where $G$ corresponds to Case ii.

Theorems & Definitions (31)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Remark 3.1
  • Lemma 3.2
  • ...and 21 more