# Characterizing nonconvex constrained best approximation using Robinson’s constraint qualification

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## 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*.

## Keywords

Lagrange multiplier Constrained best approximation Strong conical hull intersection property Robinson’s constraint qualification Perturbation property Constraint set## Notes

### Acknowledgements

The authors are very grateful to the anonymous referee and the associate editor for their useful suggestions regarding an earlier version of this paper. The comments of the referee and the associate editor were very useful and they helped us to improve the paper significantly. Also, many thanks to the associate editor for remembrance the paper [15]. This research has partially supported by Mahani Mathematical Research Center.

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