-- Antoine de Saint-Exupery
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Abstracts
Spring Conference on Analysis, April 2006
R.T. Rockafellar (University of Washington)
Generalized Second-Derivatives of Convex and Nonconvex Functions
Introduction to the subject available.Variational analysis grew out of attempts to extend concepts of convex analysis, such as subgradients and their relationship to directional derivatives, beyond convexity as well as the classical framework of differentiability. At the level of first-derivatives and their generalizations, variational analysis has been a complete success. The concepts and results are very satisfying and have turned out to have many important applications.
It is natural to ask in this light whether, at the next stage of generalized second-derivatives, comparable achievements are possible. Many interesting and significant results have in fact been obtained, but the subject is more challenging than might, at first, be expected. It does not seem possible, even for convex functions, to find a universally useful concept to build on in this direction. For instance, the classical theorem of Alexandroff about almost-everywhere differentiability of a finite convex function does not lead to a definition of generalized second-derivatives that usefully covers also the points where the convex function fails to be differentiable.
This lecture will present what is known in the subject, but also explain where the difficulties lie and where more work is needed. The essential background for such further research will be provided.
B.S. Mordukhovich (Wayne State University, Detroit, MI)
Variational analysis via the extremal principle
Related material available.We present some aspects of the geometric theory of variational analysis and its applications. This theory is mainly based on the extremal principle, which can be viewed as a variational counterpart of the classical separation principle in the case of nonconvex sets. We discuss several versions of the extremal principle, its relationships with well-recognized variational principles, and then give a number of applications. They include:
- Geometry of Banach spaces, in particular, properties of Frechet-like normals and subdifferentials; characterizations of Asplund spaces, etc.
- Comprehensive calculus rules for basic/limiting normals, subgradients, and coderivatives;
- Calculus of the so-called normal compactness properties, which are crucial in infinite-dimensional variational analysis;
- Applications to welfare economics.
J.V.Outrata (Czech Academy of Sciences, Prague)
Variational analysis in mathematical programs with equilibrium constraints
Lecture notes available.The lecture concerns the application of tools from variational analysis to a difficult optimization problem - a mathematical program with equilibrium constraint (MPEC). This problem is intrinsically nonsmooth and does not fulfill any classical constraint qualification. In particular, we will focus on
- necessary optimality conditions under general and specific constraint qualifications;
- results of the coderivative calculus needed both in optimality conditions as well as in numerical methods;
- theoretical aspects of an efficient numerical approach based on the so-called implicit programming.
Werner Roemisch (Humboldt-University, Berlin)
Stochastic programming
Lecture Notes available.An introduction to the main approaches und basic models is given: Models with probabilistic constraints and with recourse. We discuss some applications of nonsmooth and variational analysis in stochastic programming models and present some basic results on optimality, duality, stability and decomposition. We also discuss the challenges for the computational solution of stochastic programs and present some numerical experience for solving real-life problems.
J.M. Borwein (Simon Fraser University, Burnaby, Canada)
Variational Principles and Convex Applications
Lecture Notes available now!High Performance Mathematics presentation
My lectures will be built from J.M. Borwein and Qiji Zhu, Techniques of Variational Analysis, CMS/Springer-Verlag, 2005
Variational arguments connote classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse fields. The discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques.
The only broad prerequisite We anticipate is a working knowledge of undergraduate analysis and of the basic principles of functional analysis (e.g., those encountered in a typical introductory functional analysis course). The recent monograph ``Variational Analysis'' by Rockafellar and Wets has already provided an authoritative and systematical account of variational analysis in finite dimensional spaces. The forthcoming book ``Variational Analysis and Generalized Differentiation'' by Boris Mordukhovich, is a comprehensive complement to the present work."
I shall present an introduction to Variational Principles and will illustrate their use in Convex Optimization. I shall primarily use Chapters 2 and 4. An outline of Lectures follows.
- Variational Principles
- LECTURE 1
- 2.1 Ekeland Variational Principles
- LECTURE 2
- 2.2 Geometric Forms of the Variational Principle
- 2.3 Applications to Fixed Point Theorems
- 2.4 Finite Dimensional Variational Principles
- LECTURE 3
- 2.5 Borwein-Preiss Variational Principles
- Variational Techniques in Subdifferential Theory
- LECTURE 4
- 3.4 Mean Value Theorems and Applications
- Variational Techniques in Convex Analysis
- LECTURE 5
- 4.1 Convex Functions and Sets
- 4.2 Subdifferentials
- 4.3 Sandwich Theorems and Calculus
- LECTURE 6
- 4.4 Fenchel Conjugate
- 4.5 Convex Feasibility Problems
- LECTURE 7
- 4.6 Duality I7equalities for Sandwiched Functions
- 4.7 Entropy Maximization
- 6 Variational Principles in Functional Analysis
- LECTURE 8
- 6.3 Stegall Variational Principle