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 Fréchet 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.