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Abstracts

Spring School on Variational Analysis 2009

F. Bonnans

INRIA-Saclay Ile de France et CMAP
Optimal Control Problems with State Constraints

The study of optimal control problems of ODEs with state constraints leads to a very rich theory that greatly improved in recent years. The course will survey some of the most striking results and discuss important open problems.

Lecture 1: Pontryagine's principle for state constrained optimal control problems: regularity of multiplier and junction condition

Lecture 2: Possible transitions between boundary arcs and touch points

Lecture 3: The alternative optimality system

Lecture 4: The shooting algorithm

References.

J.F. Bonnans, A. Hermant: Second-order Analysis for Optimal Control Problems with Pure State Constraints and Mixed Control-State Constraints. Annales de l'I.H.P. - Nonlinear Analysis, (2008) DOI: 10.1016/j.anihpc.2007.12.002

J.F. Bonnans, A. Hermant: Well-Posedness of the Shooting Algorithm for State Constrained Optimal Control Problems with a Single Constraint and Control. SIAM J. Control Optimization 46-4(2007), p. 1398--1430.

J.F. Bonnans, A. Shapiro: Perturbation analysis of optimization problems. Springer-Verlag, Berlin, 2000.

Aris Daniilidis

Universitat Autonoma de Barcelona
Gradient Dynamical Systems, Tame Optimization and Applications

These lectures will present an introduction to what is nowadays called a Tame Optimization, with emphasis to (nonsmooth) Lojasiewicz-type inequalities and (nonsmooth) Sard-type theorems. The former topic will be introduced via the asymptotic analysis of dynamical systems of (sub)gradient type; its consequences in the algorithmic analysis (proximal algorithm, gradient-type methods) will also be discussed. The latter topic will be presented as a natural consequence of the structural assumptions made on the function (o-minimality, stratification).

These lectures aim at providing the essential background for further research. During the lectures, some open problems will be eventually mentioned.

PLAN. The lectures will be divided into three parts (approximately 2 hours each)

I. Trajectories of (sub)gradient systems

(i) Existence results and elementary properties.
(ii) Asymptotic analysis: convergence, length, Palis & DeMelo example.
(iii) Convex case: Brezis theorem, Baillon example.
(iv) Self-contracted curves. Quasiconvex planar systems.

II. Lojasiewicz inequality and generalizations

(i) Defragmented gradient curves.
(ii) The Kurdyka-Lojasiewicz inequality: characterizations and applications
(iii) Convex case: asymptotic equivalence between continuous and discrete systems.
(iv) A convex counterexample.
(v) Analytic and subanalytic case.

III. Tame Variational analysis

(i) Semialgebraic and o-minimal structures: definition, key properties.
(ii) Stratification vs Clarke subdifferential.
(iii) Sard-type theorem for (nonsmooth) tame functions.
(iv) Applications.

A. L. Dontchev

National Science Foundation, USA
Implicit Functions and Open Mappings

The classical implicit function theorem revolves around solving an equation f(p,x)=0 for x in terms of p. It is a centerpiece of mathematical analysis with countless applications, but there is much more to it than usually comes to attention. Along with the standard result, in the first part of this series of talks we will provide a rich picture of variants and alternative formulations which are important in themselves and also lay a foundation for the broader developments later in the lectures.

We will then move into that much wider territory in replacing equation-solving problems by more complicated problems for "generalized equations". Such problems arise variationally in constrained optimization, models of equilibrium, and many other areas. An important feature, in contrast to ordinary equations, is that functions obtained implicitly from their solution mappings typically lack differentiability, but often exhibit Lipschitz continuity and sometimes combine that with the existence of one-sided directional derivatives.

In the concept of a solution mapping for a problem dependent on parameters, whether formulated with equations or something more broader like variational inequalities, the possibility has always had to be faced that solutions might not exist, or might not be unique when they do exist. Even with localization of a set-valued mapping, one might be left with a set, rather than with a point, which depends on parameters, and that is anyway the usual case for the solution mappings associated with parameterized constraint systems which include inequalities. To handle these issues, the notions of Painlevé-Kuratowski convergence and Pompeiu-Hausdorff convergence for sequences of sets can be employed in developing properties of continuity and Lipschitz continuity for set-valued mappings. The Aubin property comes in as a sort of localized counterpart to Lipschitz continuity for set-valued mappings. It is tied to the concept of metric regularity, which directly relates to estimates of distances to solutions. The natural context for this is the study of how properties of a set-valued mapping correspond to properties of its set-valued inverse, or in other words, the paradigm of the inverse function theorem.

Remarkably much of the classical theory extends to set-valued mappings whose graphs are convex sets or cones instead of subspaces in infinte-dimensional spaces. Openness connects up then with metric regularity and interiority conditions on domains and ranges, as shown in the Robinson-Ursescu theorem. Infinite-dimensional inverse function theorems and open mapping theorems due to Lyusternik, Graves and Bartle-Graves can be derived and extended. Banach spaces can even be replaced to some degree by general metric spaces.

A. Ioffe

Technion - Israel Institute of Technology
Variational Analysis and Mathematical Economics

We shall consider two well known problems of mathematical economics: price equilibrium for Pareto optimal points in models of welfare economics and regularity (stability of equilibrium prices under variations of the economy parameters) in models of competitive economics, both for nonsmooth and nonconvex economies. The two problems appeal to two, in a sense oppositely extreme, settings of variational analysis. The first can be considered with extreme generality of data, practically without any a priori restrictions, the second, on the contrary, needs “reasonable” degree of nonsmoothness to make meaningful results available.

Lecture 1: A general theory of subdifferential

Brief survey of the existing subdifferential theories. An abstract concept of subdifferential. Fundamental properties of subdifferentials that allow them to work: trustworthiness, robustness, tightness. Some calculus rules. Separation theorem and extremal principle.

Lecture 2: The second fundamental theorem of welfare economics

A model of welfare economics. Pareto optimality. An abstract mathematical model. Existence of equilibrium prices.

Lecture 3: Regular and critical behavior of set-valued mappings

Quantitative characteristics (moduli) of local regularity. Subdifferential estimates. Various concepts of critical point. Stratified structures. Normal regularity and Whitney regularity. A nonsmooth extension of the Sard theorem.

Lecture 4: Regular competitive economics

A model of competitive economics. Regular economies. The classical Debreu theorem. Connection with the Thom transversality theorem. Extensions to nonsmooth economies with definable data.

 
Abstracts Spring Conference on Analysis, April 2006