Variational Analysis on Local Sharp Minima via Exact Penalization

In this paper we study local sharp minima of the nonlinear programming problem via exact penalization. Utilizing generalized differentiation tools in variational analysis such as subderivatives and regular subdifferentials, we obtain some primal and dual characterizations for a penalty function associated with the nonlinear programming problem to have a local sharp minimum. These general results are then applied to the ℓp penalty function with 0 ≤ p ≤ 1. In particular, we present primal and dual equivalent conditions in terms of the original data of the nonlinear programming problem, which guarantee that the ℓp penalty function has a local sharp minimum with a finite penalty parameter in the case of p∈(12,1] and p=12 respectively. By assuming the Guignard constraint qualification (resp. the generalized Guignard constraint qualification), we also show that a local sharp minimum of the nonlinear programming problem can be an exact local sharp minimum of the ℓp penalty function with p ∈ [0, 1] (resp. p∈[0,12]). Finally, we give some formulas for calculating the smallest penalty parameter for a penalty function to have a local sharp minimum.