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Zbl 1163.90016
Burke, James V.; Deng, Sien
Weak sharp minima revisited. III: Error bounds for differentiable convex inclusions. (English)
[J] Math. Program. 116, No. 1-2 (B), 37-56 (2009). ISSN 0025-5610; ISSN 1436-4646/e

The term weak sharp minima is coined by Ferris in the late 1980's to describe an extension of the notion of sharp minima to include the possibility of a non-unique solution set. It unifies a number of important ideas in optimization and many authors have studied this notion extensively. The notion of weak sharp minima is also an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. This is the third paper in this series. Part I of this work [Control Cybern. 31, No. 3, 439--469 (2002; Zbl 1105.90356)] provides the foundation for the theory of weak sharp minima. The basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II [Math. Program. 104, No. 2--3 (B), 235--261 (2005; Zbl 1124.90349)], the applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming are studied. In Part III, the authors continue their study of weak sharp minima by focusing on applications to error bounds for differentiable convex inclusions. A number of standard constraint qualifications for such inclusions are also examined.
[Samir Kumar Neogy (New Delhi)]

MSC 2000:: *90C25 Convex programming
90C31 Sensitivity, etc.
49J52 Nonsmooth analysis (other weak concepts of optimality)

Keywords: weak sharp minima; convex inclusion; affine convex inclusion; constraint qualification; error bounds

Citations: Zbl 1105.90356; Zbl 1124.90349

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