标题： 近端拟牛顿方法用于多目标优化问题 显示英文标题

标题： Proximal Quasi-Newton Methods for Multiobjective Optimization Problems

摘要： 我们引入了一些新的近似拟牛顿方法，用于无约束多目标优化问题（简称 UMOP），其中每个目标函数都是二阶连续可微的强凸函数与适当下半连续凸函数（但不一定可微）之和。 我们提出了适用于（UMOP）的近似 BFGS 方法、近似自缩放 BFGS 方法和近似 Huang BFGS 方法，包括有线搜索和无线搜索两种情况。 在较弱的假设下，我们证明了由这些算法生成的序列的每个聚点（如果存在的话）是（UMOP）的帕累托稳定点。 此外，我们还介绍了它们在有约束多目标优化问题和鲁棒多目标优化问题中的应用。 特别是对于鲁棒多目标优化问题，我们表明近似拟牛顿算法的子问题可以看作是带有二次不等式约束的二次最小化问题。 还进行了数值实验，以验证所提出的近似拟牛顿方法的有效性。 显示英文摘要

摘要： We introduce some new proximal quasi-Newton methods for unconstrained multiobjective optimization problems (in short, UMOP), where each objective function is the sum of a twice continuously differentiable strongly convex function and a proper lower semicontinuous convex but not necessarily differentiable function. We propose proximal BFGS method, proximal self-scaling BFGS method, and proximal Huang BFGS method for (UMOP) with both line searches and without line searches cases. Under mild assumputions, we show that each accumulation point of the sequence generated by these algorithms, if exists, is a Pareto stationary point of the (UMOP). Moreover, we present their applications in both constrained multiobjective optimization problems and robust multiobjective optimization problems. In particular, for robust multiobjective optimization problems, we show that the subproblems of proximal quasi-Newton algorithms can be regarded as quadratic minimization problems with quadratic inequality constraints. Numerical experiments are also carried out to verify the effectiveness of the proposed proximal quasi-Newton methods.