In this paper we consider the classical unconstrained nonlinear multiobjective optimization problem. For such a problem, it is particularly interesting to compute as many points as possible in an effort to approximate the so-called Pareto front. Consequently, to solve the problem we define an “a posteriori” algorithm whose generic iterate is represented by a set of points rather than by a single one. The proposed algorithm takes advantage of a linesearch with extrapolation along steepest descent directions with respect to (possibly not all of) the objective functions. The sequence of sets of points produced by the algorithm defines a set of “linked” sequences of points. We show that each linked sequence admits at least one limit point (not necessarily distinct from those obtained by other sequences) and that every limit point is Pareto-stationary. We also report numerical results on a collection of multiobjective problems that show efficiency of the proposed approach over more classical ones.

中文翻译：

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多目标优化最速下降法的收敛性

在本文中，我们考虑了经典的无约束非线性多目标优化问题。对于这样的问题，计算尽可能多的点以逼近所谓的Pareto前沿特别有趣。因此，为解决该问题，我们定义了一种“后验”算法，其泛型迭代由一组点而不是单个点表示。所提出的算法利用了相对于（可能不是全部）目标函数沿最陡下降方向外推的线搜索的优势。该算法产生的点集序列定义了一组“链接”点集。我们表明，每个链接的序列都允许至少一个极限点（不一定与其他序列获得的极限点不同），并且每个极限点都是帕累托平稳的。