基于Bregman距离的非精确邻近点算法求解拟凸多目标优化问题

An Inexact Proximal Point Method with Bregman Distance for Quasi-Convex Multiobjective Optimization

摘要: 本文针对无约束拟凸多目标优化问题，提出了一种基于Bregman距离的非精确邻近点算法，该算法在正则化项中引入Bregman距离以替代传统欧几里得距离。研究中考虑了该算法的两种误差准则(绝对误差准则和相对误差准则)，并在温和假设条件下建立收敛性理论：当目标函数连续可微时，两种准则下算法生成的序列均收敛到问题的帕累托稳定点；当目标函数真凸且下半连续时，序列收敛到问题的弱帕累托最优点。进一步地，通过引入一个额外的增长条件证明了：若正则化参数有界，采用相对误差准则的算法具有线性收敛速率；若该参数收敛至零，则可实现超线性收敛。

Abstract: This paper proposes an inexact proximal point algorithm based on Bregman distance for solving unconstrained quasi-convex multiobjective optimization problems, where the Bregman distance is employed in the regularization term. Two error criteria (absolute error criterion and relative error criterion) for the algorithm are considered. Under some mild assumptions, it is proven that: when the objective functions are continuously differentiable, the sequences generated by the algorithm under both criteria converge to the Pareto stationary points of the problem; when the objective functions are proper convex and lower semicontinuous, the sequences converge to the weak Pareto optimal points of the problem. Furthermore, by introducing an additional growth condition, it is shown that the convergence rate of the method using the relative error criterion is linear if the regularization parameters are bounded, and superlinear if these parameters converge to zero.