In this paper, we consider the set D of inequalities with nonconvex constraint functions in the face of data uncertainty. We show under a suitable condition that “perturbation property” of the robust best approximation to any \(x\in {\mathbb {R}}^n\) from the set \(\tilde{K}:={{\bar{C}}} \cap D\) is characterized by the strong conical hull intersection property (strong CHIP) of \({\bar{C}}\) and D. The set C is an open convex subset of \({\mathbb {R}}^n\) and the set D is represented by \(D:=\{x\in {\mathbb {R}}^n: g_{j}(x,v_j)\le 0, \; \forall \; v_j\in V_j, \; j=1,2,\ldots ,m\},\) where the functions \(g_j:{\mathbb {R}}^n\times V_j\longrightarrow {\mathbb {R}}, \; j=1,2,\ldots ,m,\) are continuously Fréchet differentiable that are not necessarily convex, and \(v_j\) is the uncertain parameter which belongs to an uncertainty set \(V_j\subset {\mathbb {R}}^{q_j}, \; j=1,2,\ldots ,m.\) This is done by first proving a dual cone characterization of the robust constraint set D. Finally, following the robust optimization approach, we establish Lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty under the robust nondegeneracy constraint qualification. Given examples illustrate the nature of our assumptions.

在本文中，面对数据不确定性，我们考虑具有非凸约束函数的不等式集D。我们在适当的条件下表明，对集合\（\ tilde {K}：= {{\ bar}中的任何\（x \ in {\ mathbb {R}} ^ n \）的鲁棒最佳近似的“摄动特性” {C}}} \ cap D \）的特征是\（{\ bar {C}} \）和D的强圆锥形外壳交集特性（强CHIP）。集合C是\（{\ mathbb {R}} ^ n \）的开放凸子集，集合D用\（D：= \ {x \ in {\ mathbb {R}} ^ n：g_表示。 {j}（x，v_j）\ le 0，\; \ forall \; v_j \ in V_j，\; j = 1,2，\ ldots，m \}，​​\）其中的函数\（g_j：{\ mathbb {R}} ^ n \ times V_j \ longrightarrow {\ mathbb {R}}，\; j = 1,2，\ ldots，m，\）是连续的Fréchet微分的，不一定是凸的和\（v_j \）是属于一个组的不确定性的不确定参数\（V_j \子集{\ mathbb {R}} ^ {q_j}，\; J = 1,2，\ ldots，米\。）这首先证明鲁棒约束集D的双锥特征。最后，遵循鲁棒优化方法，我们建立了鲁棒约束最佳近似的Lagrange乘子表征，该鲁棒约束最佳近似在鲁棒非退化约束条件下针对数据不确定性进行免疫。给出的例子说明了我们假设的性质。