Variational Analysis in Optimization and Control

Variational analysis has been recognized as a rapidly growing and fruitful area in mathematics concerning mainly the study of optimization and equilibrium problems, while also applying perturbation ideas and variational principles to a broad class of problems and situations that may be not of a variational nature. It can be viewed as a modern outgrowth of the classical calculus of variations, optimal control theory, and mathematical programming with the focus on perturbation/approximation techniques, sensitivity issues, and applications; see [1, 2, 3] One of the most characteristic features of modern variational analysis is the intrinsic presence of nonsmoothness, i.e., the necessity to deal with nondifferentiable functions, sets with nonsmooth boundaries, and set-valued mappings. Nonsmoothness naturally enters not only through initial data of optimization-related problems (particularly those with inequality and geometric constraints) but largely via variational principles and other optimization, approximation, and perturbation techniques applied to problems with even smooth data. In fact, many fundamental objects frequently appearing in the framework of variational analysis (e.g., the distance function, value functions in optimization and control problems, maximum and minimum functions, solution maps to perturbed constraint and variational systems, etc.) are inevitably of nonsmooth and/or set-valued structures requiring the development of new forms of analysis that involve generalized differentiation. It is important to emphasize that even the simplest and historically earliest problems of optimal control are intrinsically nonsmooth, in contrast to the classical calculus of variations. This is mainly due to pointwise constraints on control functions that often take only discrete values as in typical problems of automatic control, a primary motivation for developing optimal control theory. Optimal control has always been a major source of inspiration as well as a fruitful territory for applications of advanced methods of variational analysis and generalized differentiation. In this talk we discuss some new trends and developments in variational analysis and its applications mostly based on the author’s recent 2-volume book [1, 2]. Generalized differentiation lies at the heart of variational analysis and its applications. We systematically develop a geometric dual-space approach to generalized differentiation theory revolving around the extremal principle, which can be viewed as a local variational counterpart of the classical convex separation in nonconvex settings. This principle allows us to deal with nonconvex derivative-like constructions for sets (normal cones), set-valued mappings (coderivatives), and extended-real-valued functions (subdifferentials). These constructions are defined directly in dual spaces and, being nonconvex-valued, cannot be generated by any derivative-like constructions in primal spaces (like tangent cones and directional derivatives). Nevertheless, our basic nonconvex constructions enjoy comprehensive/full calculus, which happens to be significantly better than those available for their primal and/or convexvalued counterparts. The developed generalized differential calculus based on variational