Characterization of Strong Stability of Stationary Solutions of Nonlinear Programs with a Finite Number of Equality Constraints and an Abstract Convex Constraint

In this report we treat nonlinear programs Pro(f, h;K) having an objective function f , a finite number of equality constraints h(x) = (h1(x), · · · , hl(x)) = 0, and an abstract convex constraint x ∈ K with its convex set K. Our particular interest is an algebraic criterion for a locally isolated stationary solution to be strong stable, in the sense of Kojima, under a Linear Independence Constraint Qualification condition defined to those programs. First, we introduce a simple sufficient condition for semismoothness of the Euclidean projector ρ+K onto K. Semismoothness of the Euclidean projectors onto closed convex cones pointed at 0 follows directly from this sufficiency. Secondly, under the condition of semismoothness of ρ+K and what we call the regular boundary condition for K, we characterize strong stability of a locally isolated stationary solution x̄, with (x̄, λ̄) its associate stationary point, in terms of B-subderivative ∂Bψ(x̄, λ̄; f, h) of some appropriately defined map ψ(x, λ; f, h) for programs Pro(f, h;K). This result is a generalization of the theory that Kojima developed in his famous paper. Thirdly, we state an explicit formula for the Jacobian of the Euclidean projector onto any closed convex set satisfying the C stratification, and interpret the regular boundary condition in terms of principal curvatures of the stratum.