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Weak sharp minima revisited Part I : basic theory

Burke J., Deng S.

Control and Cybernetics

2002

Vol. 31, no 3

439-469

The notion of sharp minima, or strongly unique local minima, emerged in the late 1970's as 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. The work of Cromme and Polyak is of particular importance in this development. In the late 1980's Ferris coined the term weak sharp minima to describe the extension of the notion of sharp minima to include the possibility of a non-unique solution set. This notion was later extensively studied by many authors. Of particular note in this regard is the paper by Burke and Ferrris which gives an extensive exposition of the notion and its impact on convex programming and convergence analysis in finite dimensions. In this paper we build on the work of Burke and Ferris. Specifically, we generalize their work to the normed linear space setting, further dissect the normal cone inclusion characterization for weak sharp minima, study the asymptotic properties of weak sharp minima in terms of associated recession functions, and give new characterizations for local weak sharp minima and boundely weak sharp minima. This paper is the first of a two part work on this subject. In Part II, we study the links between the notions of weak sharp minima, bounded linear regularity, linear regularity, metric regularity, and error bounds in convex programming. Along the way, we obtain both new results and reproduce many existing results from a fresh perspective.