In this paper, we first employ the subdifferential closedness condition and Guignard’s constraint qualification to present “dual cone characterizations” of the constraint set \( \varOmega \) 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 \(x \in {\mathcal {H}}\) from the convex set \( \tilde{\varOmega }:=C \cap \varOmega \) by using the strong CHIP of \(\lbrace C,\varOmega \rbrace ,\) where C is a non-empty closed convex set in the Hilbert space \({\mathcal {H}}.\) 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.

在本文中，我们首先利用微分封闭性条件和Guignard的约束条件来表示具有无限非凸不等式约束的约束集\（\ varOmega \）的“双锥刻画” ，其中约束函数是Fréchet微分的，不一定是凸的。接下来，我们提供“强圆锥形船体相交特性”（强CHIP）成立的充分条件，此外，我们建立必要和充分条件，以表征对任何\（x \ in {\ mathcal凸集\（\ tilde {\ varOmega}：= C \ cap \ varOmega \）中的{H}} \）\（\ lbrace C，\ varOmega \ rbrace，\）其中Ç是在希尔伯特空间中的非空闭凸集\（{\ mathcal {H}}。\）最后，我们得出的“拉格朗日乘数表征”次微分封闭性条件和吉格纳约束条件下的约束最佳逼近。提出了几个说明性的例子来阐明我们的结果。