A Partially Derivative-Free Proximal Method for Composite Multiobjective Optimization in the Hölder Setting

TL;DR

The paper addresses composite multiobjective optimization where each objective is the sum of a differentiable part and a convex, possibly nondifferentiable part. It introduces a partially derivative-free proximal method (PDFPM) that builds a quadratic surrogate using gradient approximations and a proximal term, then adaptively updates a penalty parameter to secure descent. Under Hölder-gradient assumptions, the authors prove a finite-iteration complexity bound of $O(ε^{-(β+1)/β})$ to reach an $ε$-approximate Pareto point and demonstrate the method's effectiveness through robust numerical experiments and comparisons with ProxGrad and CondG. The work highlights a flexible framework that accommodates mixed smoothness across objectives, does not require explicit Lipschitz constants, and remains robust under uncertainty, with practical implications for large-scale, non-smooth, multiobjective problems.

Abstract

This paper presents an algorithm for solving multiobjective optimization problems involving composite functions, where we minimize a quadratic model that approximates $F(x) - F(x^k)$ and that can be derivative-free. We establish theoretical assumptions about the component functions of the composition and provide comprehensive convergence and complexity analysis. Specifically, we prove that the proposed method converges to a weakly $\varepsilon$-approximate Pareto point in at most $\mathcal{O}\left(\varepsilon^{-\frac{β+1}β}\right)$ iterations, where $β$ denotes the Hölder exponent of the gradient. The algorithm incorporates gradient approximations and a scaling matrix $B_k$ to achieve an optimal balance between computational accuracy and efficiency. Numerical experiments on a collection of benchmark problems are presented, illustrating the practical behavior of the proposed approach and its competitiveness with existing composite algorithms.

Figures (5)

• Figure 1: Objective space and approximated Pareto front obtained by the PDFPM for the AAS1.
• Figure 2: Objective space and approximated Pareto front obtained by the PDFPM for the AAS2.
• Figure 3: Comparison between the complete objective space and the Pareto front approximations obtained by PDFPM for the AAS1 and AAS2 problems under different uncertainty levels.
• Figure 4: Performance profiles with respect to the number of function and gradient evaluations.
• Figure 5: Performance profiles with respect to the number of iterations and the total execution time.

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Lemma 3.1
• proof
• Remark 3.4
• Example 3.1
• Lemma 3.2