Mathematical Sciences: RUI: Set Convergence, Convex Analysis, and Optimization

比尔教授一直从事赋范线性空间闭凸子集的拓扑研究。这样的拓扑结构产生凸函数上的拓扑结构，其中一个函数与它的铭文，即它的图上或上面的点的集合一致。在过去的两年里，Beer教授引入了一个与凸集序列的Mosco收敛相容的拓扑，证明了当且仅当底层空间是自反的，它对对偶是稳定的。得到了该拓扑的若干表征，研究了凸集上各种算子和泛函的连续性性质。针对这种Mosco拓扑在非自反情况下的不适宜性，并考虑到先前关于线性泛函及其水平集范数收敛的一些有希望的结果，Beer教授研究了距离泛函在有界集合上一致收敛的拓扑。这种拓扑不仅在对偶性方面不受限制地稳定，而且在凸优化问题解的稳定性方面也是正确的收敛概念。Beer教授将继续研究凸集的收敛性，特别关注与巴拿赫空间几何的联系、多函数的近似、特殊运算和泛函。他还希望参与集收敛在非光滑分析基础的发展研究中的应用，例如某些优化问题。

Professor Beer has been involved in the study of topologies on the closed convex subsets of a normed linear space. Such topologies give rise to topologies on convex functions, with a function identified with its epigraph, the set of points on or above its graph. In the last two years, Professor Beer introduced a topology compatible with Mosco convergence of sequences of convex sets, showing that it is stable with respect to duality if and only if the underlying space is reflexive. Several characterizations of the topology were obtained, and continuity properties of a variety of operators and functionals on convex sets were studied. In response to the unsuitability of this Mosco topology in the nonreflexive case, and in view of some prior promising results regarding the norm convergence of linear functionals and their level sets, Professor Beer studied the topology of uniform convergence of distance functionals on bounded sets. Not only is this topology stable with respect to duality without restriction, but it is also the right convergence notion in terms of the stability of solutions of convex optimization problems. Professor Beer will continue his studies on convergence of convex sets, with particular attention to connections with Banach space geometry, approximation of multifunctions, and specific operations and functionals. He also hopes to become involved in applications of set convergence in evolving research in the foundations of nonsmooth analysis, e.g., to certain optimization problems.