arXiv 2210.095962022-10-18math.FAmath.OC

凸集生成空间理论及其在巴拿赫空间凸规划可解性中的应用

Theory of generating spaces of convex sets and their applications to solvability of convex programs in Banach spaces

Lixin Cheng, Weihao Mao

摘要：

当优化理论家考虑无限维空间中的最优化问题时，需要处理闭凸子集（通常是锥），而这些闭凸子集的内部大多是空的。这些子集经常妨碍优化理论家应用强大的技术来研究这些最优化问题。本文利用非支撑点给出了巴拿赫空间与其凸闭子集的非支撑点相关的生成空间。然后，对于无限维空间中的最优化问题，在某些一般情况下，我们用包含解的生成空间代替原空间。因此，这种方法使我们能够将强大的经典技术应用于非常一般的无穷维空间类中的最优化问题。基于泛函分析，从经典的巴拿赫空间到可分的巴拿赫空间，从巴拿赫格到格化，给出了生成空间的特征，证明了生成空间实际上线性等距于$L_\infty$（$\ell _\infty$）或其闭子空间.因此，这些技术中涉及的连续线性泛函可以从$L_\infty^*$（$\ell_\infty^*$）中选择。在此基础上，进一步研究了该方法在罚原理、Lagrange对偶和标量化函数中的应用。

英文摘要：

When optimization theorists consider optimization problems in infinite dimensional spaces, they need to deal with closed convex subsets(usually cones) which mostly have empty interior. These subsets often prevent optimization theorists from applying powerful techniques to study these optimization problems. In this paper, by nonsupport point, we present generating spaces which are relative to a Banach space and a nonsupport point of its convex closed subset. Then for optimization problems in infinite dimensional spaces, in some general cases, we replace original spaces by generating spaces while containing solutions. Thus this method enable us to apply powerful classical techniques to optimization problems in very general class of infinite dimensional spaces. Based on functional analysis, from classical Banach spaces to separable Banach spaces, from Banach lattice to latticization, we give characterizations of generating spaces and conclude that they are actually linearly isometric to $L_\infty$($\ell _\infty$) or their closed subspaces. Thus continuous linear functional involved in these techniques could be chosen from $L_\infty^*$($\ell_\infty^*$). After that, applications in Penalty principle, Lagrange duality and scalarization function are further studied by this method.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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