Variational Analysis,Variational Geometry,and Optimization

报告题目:Variational Analysis, Variational Geometry, and Optimization

报 告 人:Prof. R. T. Rockafellar

报告时间:2019/4/10 10:00:00


专家简介:西雅图华盛顿大学数学和应用数学系的首席教授,主要从事优化研究,研究方向是凸分析和变分分析,重点是应用于随机规划、最优控制、经济、金融和工程。曾获约翰•冯•诺依曼(John von Neumann Theory Prize ),弗雷德里克•兰彻斯特奖(Frederick W. Lanchester Prize)、丹茨格奖(Dantzig Prize)等多项奖励。现为多个国际顶级期刊主编,编辑和编委。在国际优化控制权威期刊上发表大量学术论文,出版专著6部,其中专著《凸分析》和《变分分析》是从事非线性分析和优化理论学者的必读的经典教材。

报告摘要:Classical analysis covers many function-constructing operations, such as addition, multiplication, composition and even integration, but not the operations that are essential in optimization, namely minimization and maximization. The simple reason is that those operations don't preserve differentiability.  If a function g(x) is defined as the minimum (or the maximum) of f(x,y) with respect to y in some set Y, no amount of differentiability of f in x and y will carry over, in general, to g being differentiable in x.    

Variational analysis, as an extension of classical analysis which encompasses also convex analysis, gets around this by introducing one-sided concepts of generalized differentiability which moreover have a basis in set convergence very different from the usual pointwise convergence of different quotients.  Variational geometry provides powerful support by associating one-sided tangent and normal "cones" instead of subspaces to the points of a set in a linear space. Novel concepts of regularity,unanticipated in classical theory, then come up.  

This talk will aim at explaining these ideas on broad introductory level.