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# Local minimizers of semi-algebraic functions

2019-06-29

Title: Local minimizers of semi-algebraic functions

Reporter: Doc. Tien-Son PHAM (University of Vietnam)

Time: June 1, 2019 (Monday) PM 15:00-16:00

Location: A1101# room, Innovation Park Building

Contact: A. Prof. GUO Feng (tel:84708351-8088)

Abstract: Consider a semi-algebraic function f: R^N->R, which is continuous around a point \bar{x}. Using the so-called tangency variety of f at \bar{x}, we first provide necessary and sufficient conditions for \bar{x} to be a local minimizer of f, and then in the case where \bar{x} is an isolated local minimizer of f, we define a tangency exponent'' \alpha_*>0 so that for any \alpha the following four conditions are always equivalent:

1) the inequality \alpha>\alpha_* holds;

2) the point \bar{x} is an \alpha-order sharp local minimizer of f

3) the limiting subdifferential \partial f of f is (\alpha-1)-order strongly metrically subregular at \bar{x} for 0; and

4) the function f satisfies the Lojaseiwcz gradient inequality at \bar{x} with the exponent 1-1/(\alpha).

Besides, we also present a counterexample to a conjecture posed by Drusvyatskiy and Ioffe in [Math. Program. Ser. A, 153(2):635--653, 2015].

The brief introduction to the reporter: Professor Tien-Son PHAM's main research fields are singularity theory and polynomial optimization. He has done a lot of important work on the problems of Qojasiewicz inequality, generalized key values of polynomial functions and relaxation of semi-definite programming for polynomial optimization. He has published nearly 60 papers and one monograph in international authoritative journals, including SAM Journal on Optimization, Mathematical Prog. More than 10 papers have been published in excellent magazines such as ramming.