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

比尔教授一直从事拓扑学的研究 关于赋范线性空间的闭凸子集。 等 拓扑在凸函数上产生拓扑， 用其上图标识的函数，或 在它的曲线图上。 在过去的两年里，比尔教授 引入了一个拓扑兼容的Mosco收敛 序列的凸集，表明它是稳定的关于 当且仅当底层空间是自反的。 得到了拓扑的几个特征， 各种算子和泛函的连续性 研究了凸集的性质。 针对不适合 这个Mosco拓扑在非自反的情况下，并鉴于一些 先前关于线性范数收敛的有希望的结果 泛函及其水平集，Beer教授研究了 上距离泛函的一致收敛拓扑 有界集合 这个拓扑结构不仅相对于 对偶性没有限制，但它也是右收敛 凸函数解的稳定性概念 优化问题 比尔教授将继续他的研究收敛 凸集，特别注意与巴拿赫的连接 空间几何，多函数逼近，以及具体的 操作和函数。 他还希望参与 集合收敛在进化研究中的应用 非光滑分析的基础，例如，到一定的优化 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.