• Variational Analysis and Metric Regularity Theory
  • Speaker: Prof Emeritus Alexander Ioffe, Technion
  • Title: Variational Analysis and Metric Regularity Theory
  • Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
  • Access Grid Venue: CARMA [ENQUIRIES]
  • Time and Date: 2:00 pm, Tue, 16th Jul 2013
  • Download: Variational analysis and metric regularity theory (248K)
  • CARMA is offering this short course, to be given by Professor Emeritus Alexander Ioffe, Technion, Israel, via the Access Grid, comprising of three two-hour lectures over four days. Details can be seen on our events webpage, and the course flyer is available here. (Please note that the Wednesday session has moved to Thursday.)
  • Dates and times:
    • Tuesday, 16th July 2:00-4:00 pm (AEST)
    • Thursday, 18th July 2:00-4:00 pm (AEST)
    • Friday, 19th July 2:00-4:00 pm (AEST)
  • Abstract:

    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:

    1. Classical theory;
    2. Metric "phenomenological" theory;
    3. Metric infinitesimal theory (with the concept of "slope" of DeGiorgi-Marino-Tosques at the centre);
    4. Banach theory (with subdifferentials and coderivatives as the main instrument of analysis);
    5. 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:
    1. Metric fixed point theory (with emphasis on two mappings models, e.g. $F:X \to Y$ and $G:Y \to X$);
    2. Subregularity, exact penalties and general approach to necessary optimality conditions (optimality alternative);
    3. Stability of solutions of systems of convex inequalities;
    4. Curves of steepest descent for non-differentiable functions;
    5. Von Neumann's method of alternate projections for nonconvex sets;
    6. Tame optimization and generically good behaviour;
    7. 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.

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