The paper is concerned with multiobjective sparse optimization problems, i.e. the problem of simultaneously optimizing several objective functions and where one of these functions is the number of the non-zero components (or the \(\ell _0\)-norm) of the solution. We propose to deal with the \(\ell _0\)-norm by means of concave approximations depending on a smoothing parameter. We state some equivalence results between the original nonsmooth problem and the smooth approximated problem. We are thus able to define an algorithm aimed to find sparse solutions and based on the steepest descent framework for smooth multiobjective optimization. The numerical results obtained on a classical application in portfolio selection and comparison with existing codes show the effectiveness of the proposed approach.

中文翻译：

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基于凹面优化的稀疏多目标规划方法

本文涉及多目标稀疏优化问题，即同时优化多个目标函数的问题，其中这些函数之一是解的非零分量（或\（\ ell _0 \）-范数）的数量。我们建议处理\（\ ell _0 \）-根据平滑参数通过凹近似逼近。我们陈述了原始非光滑问题和光滑近似问题之间的一些等价结果。因此，我们能够定义一种算法，该算法旨在找到稀疏解，并基于最陡峭的下降框架进行平滑的多目标优化。在投资组合选择的经典应用中获得的数值结果以及与现有代码的比较表明了该方法的有效性。