Lagrange Multiplier Characterizations of Constrained Best Approximation with Infinite Constraints

In this paper, we first employ the subdifferential closedness condition and Guignard’s constraint qualification to present “dual cone characterizations” of the constraint set $$ \varOmega $$ Ω with infinite nonconvex inequality constraints, where the constraint functions are Fréchet differentiable that are not necessarily convex. We next provide sufficient conditions for which the “strong conical hull intersection property” (strong CHIP) holds, and moreover, we establish necessary and sufficient conditions for characterizing “perturbation property” of the best approximation to any $$x \in {\mathcal {H}}$$ x ∈ H from the convex set $$ \tilde{\varOmega }:=C \cap \varOmega $$ Ω ~ : = C ∩ Ω by using the strong CHIP of $$\lbrace C,\varOmega \rbrace ,$$ { C , Ω } , where C is a non-empty closed convex set in the Hilbert space $${\mathcal {H}}.$$ H . Finally, we derive the “Lagrange multiplier characterizations” of constrained best approximation under the subdifferential closedness condition and Guignard’s constraint qualification. Several illustrative examples are presented to clarify our results.