Lagrange Multiplier Characterizations of Robust Best Approximations under Constraint Data Uncertainty∗

In this paper we explain how to characterize the best approximation to any x in a Hilbert space X from the set C ∩ {x ∈ X : gi(x) ≤ 0, i = 1, 2, · · · ,m} in the face of data uncertainty in the convex constraints, gi(x) ≤ 0, i = 1, 2, · · · ,m, where C is a closed convex subset of X. Following the robust optimization approach, we establish Lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty. This is done by characterizing the best approximation to any x from the robust counterpart of the constraints where the constraints are satisfied for all possible uncertainties within the prescribed uncertainty sets. Unlike the traditional Lagrange multiplier characterizations without data uncertainty, for constrained best approximation problems in the face uncertainty, we show that the strong conical hull intersection property (strong CHIP) alone is not sufficient to guarantee the Lagrange multiplier characterizations. We present conditions which guarantee that the strong CHIP is necessary and sufficient for the multiplier characterization. We also establish that the strong CHIP is automatically satisfied for the cases of polyhedral constraints with polytope uncertainty, and linear constraints with interval uncertainty. As an application, we show how robust solutions of shape preserving interpolation problems under ellipsoidal and box uncertainty cases can be obtained in terms of Lagrange multipliers under strict robust feasibility conditions.