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

Beer教授一直致力于赋范线性空间的闭凸子集上的拓扑研究。这样的拓扑产生了凸函数上的拓扑，一个函数用它的上图来标识，即它的图上或图上的点的集合。在过去的两年里，Beer教授引入了一种与凸集序列的Mosco收敛相容的拓扑，证明了它关于对偶稳定的充要条件是基础空间是自反的。得到了该拓扑的几个刻画，并研究了凸集上各种算子和泛函的连续性性质。鉴于这种MOSCO拓扑在非自反情形下的不适性，以及前人关于线性泛函及其水平集的范数收敛的一些有希望的结果，Beer教授研究了有界集上距离泛函的一致收敛的拓扑。这种拓扑不仅不受限制地相对于对偶稳定，而且在凸优化问题解的稳定性方面也是正确的收敛概念。比尔教授将继续研究凸集的收敛，特别关注与Banach空间几何的联系，多功能函数的逼近，以及特定的运算和泛函。他还希望在非光滑分析的基础上，参与集合收敛在进化研究中的应用，例如，对某些优化问题。

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.