ABSTRACT
In the preceding chapters we studied the necessary and sufficient optimality conditions for x¯ ∈ Rn to be a point of minimizer for the convex optimization problem wherein a convex objective function f is minimized over a convex feasible set C ⊂ Rn. From Theorem 2.90, if the objective function f is strictly convex, then the point of minimizer x¯ is unique. The notion of unique minimizer was extended to the concept of sharp minimum or, equivalently, strongly unique local minimum. The ideas of sharp minimizer and strongly unique minimizer were introduced by Polyak [94, 95] and Cromme [29]. These notions played an important role in the approximation theory or the study of perturbation in optimization problems and also in the analysis of the convergence of algorithms [1, 26, 56]. Below we define the notion of sharp minimum.
