Smoothing Accelerated Proximal Gradient Method with Backtracking for Nonsmooth Multiobjective Optimization
Huang Chengzhi
TL;DR
This work tackles nonsmooth composite multiobjective optimization where each objective is $F_i(x)=f_i(x)+g_i(x)$ with $f_i$ convex but possibly nonsmooth. It introduces a smoothing framework and a backtracking-based smoothing accelerated proximal gradient method with extrapolation (SAPGM) that adaptively tunes a local Lipschitz estimate and smoothing parameter to ensure convergence to weak Pareto optima. Theoretical results establish convergence rates that depend on a parameter $\sigma\in(0,2)$, yielding rates like $u_0(x_{k+1})=O(k^{-\sigma})$ (or analogous) and providing key auxiliary estimates that guarantee monotone progress. Numerical experiments show SAPGM significantly outperforms subgradient methods and the DNNM baseline in runtime, iterations, and function evaluations, validating its practical efficiency for nonsmooth multiobjective problems.
Abstract
For the composite multi-objective optimization problem composed of two nonsmooth terms, a smoothing method is used to overcome the nonsmoothness of the objective function, making the objective function contain at most one nonsmooth term. Then, inspired by the design idea of the aforementioned backtracking strategy, an update rule is proposed by constructing a relationship between an estimation sequence of the Lipschitz constant and a smoothing factor, which results in a backtracking strategy suitable for this problem, allowing the estimation sequence to be updated in a non-increasing manner. On this basis, a smoothing accelerated proximal gradient algorithm based on the backtracking strategy is further proposed. Under appropriate conditions, it is proven that all accumulation points of the sequence generated by this algorithm are weak Pareto optimal solutions. Additionally, the convergence rate of the algorithm under different parameters is established using a utility function. Numerical experiments show that, compared with the subgradient algorithm, the proposed algorithm demonstrates significant advantages in terms of runtime, iteration count, and function evaluations.