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Weak sharp minima in mathematical programming. (English) Zbl 0791.90040
Summary: The notion of a sharp, or strongly unique, minimum is extended to include the possibility of a nonunique solution set. These minima will be called weak sharp minima. Conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases. These conditions are also shown to be sufficient under the appropriate convexity hypotheses. The existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem. In particular, a result of Mangasarian and Meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty. Consequences for the convergence theory of algorithms are also examined, especially conditions yielding finite termination.

MSC:
90C20Quadratic programming
90C30Nonlinear programming
65K05Mathematical programming (numerical methods)
90C25Convex programming
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)