Robust constrained best approximation with nonconvex constraints

Abstract

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.