Type: Seminar

Modified: Fri 31 May 2013 1818; Mon 15 Jul 2013 1438

Distribution: World

Expiry: 19 Jul 2013

CalTitle1: AGR Course: Variational analysis and metric regularity theory

Calendar2: 18 Jul 2013 1400-1600

CalLoc2: AGR Seminar

CalTitle2: AGR Course: Variational analysis and metric regularity theory

Calendar3: 19 Jul 2013 1400-1600

CalLoc3: AGR Seminar

CalTitle3: AGR Course: Variational analysis and metric regularity theory

Auth: mathas@pmathas.pc (assumed)

**Host venue**

University of Newcastle (CARMA)

**Abstract**

The University of Newcastle is offering a short course in the period 16 - 19 July 2013, entitled Variational analysis and metric regularity theory. The course is open to postgraduate students, honours students and researchers who have access to Access Grid facilities.

If you are interested in participating, please register by sending an email to agr@amsi.org.au by 30 June with the following information: your full name, email address, University affiliation, and postal address.Overview of Course Content The classical regularity theory is centred around the implicit and Lyusternik-Graves theorems, on the one hand, and the Sard theorem and transversality theory, on the other. The theory (and a number of its applications to various problems of variational analysis) to be discussed in the course deals with similar problems for non-differentiable and set-valued mappings. This theory grew out of demands that came from needs of (mainly) optimization theory and subsequent understanding that some key ideas of the classical theory can be naturally expressed in purely metric terms without mention of any linear and/or differentiable structures.

**Topics to be covered**

The "theory" part of the course consists of five sections:

- Classical theory;
- Metric "phenomenological" theory;
- Metric infinitesimal theory (with the concept of "slope" of DeGiorgi-Marino-Tosques at the centre);
- Banach theory (with subdifferentials and coderivatives as the main instrument of analysis);
- Finite dimensional theory (mainly mappings with special structures, e.g. semi-algebraic, and non-smooth extensions of the Sard theorem and transversality theory). In the second part of the course (some or all of) the following applications will be discussed:
- Metric fixed point theory (with emphasis on two mappings models, e.g. F:X\to Y and G:Y\to X);
- Subregularity, exact penalties and general approach to necessary optimality conditions (optimality alternative);
- Stability of solutions of systems of convex inequalities;
- Curves of steepest descent for non-differentiable functions;
- Von Neumann's method of alternate projections for nonconvex sets;
- Tame optimization and generically good behaviour;
- Mathematical economics: extension of Debreu's stability theorem for non-convex and non-smooth utilities.

Formally, for understanding of the course basic knowledge of functional analysis plus some acquaintance with convex analysis and nonlinear analysis in Banach spaces (e.g. Frechet and Gateaux derivatives, implicit function theorem) will be sufficient. Understanding of the interplay between analytic and geometric concepts would be very helpful.

**Contact person at Newcastle**

Prof. Jon Borwein

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If you would like to attend this course in our access grid room then please check to see if the grid is already booked at this time and send an email to accessgridroom@maths.usyd.edu.au to let the CSOs know that you would like to attend.