Characterizing nonconvex constrained best approximation using Robinson’s constraint qualification

Abstract

Extending a powerful fundamental result of constrained best approximation, we show under a suitable condition that the “perturbation property” of the best approximation $$x_0$$ to any $$x \in {{\mathbb {R}}}^n$$ from a convex set $${\tilde{K}}:=C \cap K$$ is characterized by the strong conical hull intersection property (CHIP) of C and K at $$x_0$$ . The set $$C \subseteq {{\mathbb {R}}}^n$$ is closed and convex and the set K has the representation that $$K:=\{x\in {{\mathbb {R}}}^n : -g(x) \in S \}$$ , where the function $$g: {{\mathbb {R}}}^n \longrightarrow {{\mathbb {R}}}^m$$ is continuously Frechet differentiable that is not necessarily convex. We prove this by first establishing a dual cone characterization of the constraint set K. Our results show that the convex geometry of $${\tilde{K}}$$ is critical for the characterization rather than the representation of K by convex inequalities, which is commonly assumed for the problems of best approximation from a convex set. In the special case where the set K is convex, we show that the Lagrange multiplier characterization of best approximation holds under the Robinson’s constraint qualification. The lack of representation of K by convex inequalities is supplemented by the Robinson’s constraint qualification, but the characterization, even in this special case, allows applications to problems with $$g:=(g_1, g_2, \ldots , g_m)$$ , where $$g_1, g_2, \ldots , g_m$$ are quasi-convex functions, as it guarantees the convexity of K.