This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract setting, the set of Pareto optimal solutions of any multiobjective optimization problem. We then provide sufficient topological conditions to ensure the existence of Pareto optimal solutions. Next, we determine the Pareto optimal solutions of convex max–min problems involving continuous linear operators defined on Banach spaces. We prove that the set of Pareto optimal solutions of a convex max–min of form $max\parallel T\left(x\right)\parallel$, $min\parallel x\parallel$ coincides with the set of multiples of supporting vectors of T. Lastly, we apply this result to convex max–min problems in the Hilbert space setting, which also applies to convex max–min problems that arise in the design of truly optimal coils in engineering.

中文翻译：

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连续线性算子多重优化的帕累托最优

该手稿确定了涉及在Banach空间和Hilbert空间上定义的连续线性算子的某些多目标优化问题的Pareto最优解集。这些多重优化问题通常在工程中出现。为了实现我们的目标，我们首先在抽象的背景下描述了任何多目标优化问题的帕累托最优解的集合。然后，我们提供足够的拓扑条件，以确保存在Pareto最优解。接下来，我们确定涉及在Banach空间上定义的连续线性算子的凸最大-最小问题的Pareto最优解。我们证明了凸最大-最小形式的帕累托最优解的集合$最大限度\parallel Ť（X）\parallel$， $分\parallel X\parallel$与T的支持向量的倍数集重合。最后，我们将此结果应用于希尔伯特空间设置中的最大凸-最小问题，也将其应用于工程中真正最佳线圈的设计中出现的最大凸-最小问题。