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The exponent of Hoelder calmness for level sets of polynomials

Since solutions of optimization problems may often only be found approximately, it is usefull to think about their (local) stability, i.e. to analyze the behaviour of the set of solutions under perturbations. In particular this plays a role in multilevel programs where one uses preliminary results to solve another problem. One tool for such analysis is the stability property called calmness, which can be generalized to some Hoelder type characteristic. Hoelder calmness of (nonempty) solution sets of inequality systems may be described in terms of (local) error bounds. In 1952 Hoffman showed that for (finite) systems of affine functions there is an error bound which yields calmness of the solution set. Using the Hoermander-Lojasiewicz inequality Luo/Luo and Luo/Pang gave a proof of Hoelder calmness for systems of polynomials and even analytic functions. But since the Hoermander-Lojasiewicz inequality is based on the Tarski-Seidenberg principle one only knows that there is an exponent for Hoelder calmness but cannot say more about it. Here we will show that for solution sets of one quadratic polynomial the exponent is one-half and give some ideas clarifying the correlation between the degree of the polynomials defining some set and the exponent of Hoelder calmness.
Type of Seminar:
Optimization and Applications Seminar
Jan Heerda
Humboldt University Berlin
Apr 12, 2010   16:30

HG G 19.1, Rämistrasse 101
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