Approximate Solutions for Abstract Inequality Systems

We consider conic inequality systems of the type F (x) ≥K 0, with approximate solution x0 associated to a parameter τ , where F is a twice Fréchet differentiable function between Hilbert spaces X and Y , and ≥K is the partial order in Y defined by a nonempty convex (not necessarily closed) cone K ⊆ Y . We prove that, under the suitable conditions, the system F (x) ≥K 0 is solvable, and the ratio of the distance from x0 to the solution set S over the distance from F (x0) to the cone K has an upper bound given explicitly in terms of τ and x0. We show that the upper bound is sharp. Applications to analytic function inequality/equality systems on Euclidean spaces are given, and the corresponding results of Dedieu [SIAM J. Optim., 11 (2000), pp. 411–425] are extended and significantly improved.