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On inexact projected gradient methods for solving variable vector optimization problems
Optimization and Engineering ( IF 2.1 ) Pub Date : 2020-11-25 , DOI: 10.1007/s11081-020-09579-8
J. Y. Bello-Cruz , G. Bouza Allende

Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, inexact projected gradient methods for solving smooth constrained vector optimization problems on variable ordered spaces are presented. It is shown that every accumulation point of the generated sequences satisfies the first-order necessary optimality condition. Moreover, under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences are weakly efficient points. The convergence results are also derived in the particular case in which the problem is unconstrained and even if inexact directions are taken as descent directions. Furthermore, we investigate the application of the proposed method to optimization models where the domain of the variable order map coincides with the image of the objective function. In this case, similar concepts and convergence results are presented. Finally, some computational experiments designed to illustrate the behavior of the proposed inexact methods versus the exact ones (in terms of CPU time) are performed.



中文翻译:

关于求解变量向量优化问题的不精确投影梯度方法

可变顺序结构可对两点之间的比较取决于点到圆锥图的情况进行建模。本文提出了不精确的投影梯度法来解决变序空间上的光滑约束向量优化问题。结果表明,所生成序列的每个累加点均满足一阶必要最优性条件。此外,在针对目标函数的合适凸假设下,证明了任何生成序列的所有累加点都是弱有效点。在问题不受约束并且即使将不精确的方向视为下降方向的特定情况下,也可以得出收敛结果。此外,我们研究了该方法在变量阶图的域与目标函数的图像重合的优化模型中的应用。在这种情况下,将给出相似的概念和收敛结果。最后,进行了一些计算实验,这些实验旨在说明所提出的不精确方法与精确方法(就CPU时间而言)的行为。

更新日期:2020-11-25
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