Some aspects of Variational Analysis and Applications
Organizer:
Vladimir Goncharov
Abstract:
The goal of the mini Symposium is to bring together various tools of Variational Analysis, modern discipline, which partially unifies such traditional fields of mathematics
as Calculus of Variations, Optimal Control, Games Theory, Nonsmooth and Multivalued Analysis, etc. The crucial term in this discipline is variation, which serves mostly
for optimization or for the choice of an optimal strategy, optimal values of some parameters needed, e.g., to improve the quality of technological processes, of economic
behaviour and so on.
Theoretical results in Variational Analysis are often obtained through so named variational as well as optimality principles. The First ones, such as Ekeland's Variational
Principle, Strong Maximum Principle for elliptic systems, etc., being the fundamental mathematical properties, permit to prove existence, uniqueness and regularity of
minimizers, or to study their qualitative characteristics; while others (e.g., the famous
Pontryagin's Maximum Principle or Bellman's Dynamic Programming Principle) give
necessary (sometimes sucient) conditions of minima (maxima) that is basis for the
development of effective numerical methods and algorithms.
On the other hand, various optimization problems appearing in practice (such as variational or optimal control problems with phase constraints) require detailed studying
of new classes of sets and functions, not necessarily convex or smooth. The results
obtained in this direction being of big importance themselves can be successfully applied
in optimization and in various fields of Analysis.
Thus, the mini Symposium focuses mainly on both variational and optimality principles emphasized above and recent achievements in studying of variational (geometric)
properties of sets and functions. Among the concrete theoretical subjects covered by
mini Symposium's talks let us point out the following:
the weak convexity and proximal regularity of closed sets in Hilbert and Banach
spaces;
controllable sweeping processes;
existence and qualitative properties (e.g., a priori estimates) of minimizers in
variational problems;
optimality conditions for the problems governed by differential inclusions.
