Approximate solutions for abstract inequality systems (Q2848193)

Given a twice Fréchet differentiable function \(F:X\to Y\), where \(X,Y\) are Hilbert spaces, and a partial order \(\geq_K\) in \(Y\) defined via a nonempty convex cone \(K\subseteq Y\), the conic inequality system \(F(x)\geq_K 0\) is considered. Under the assumption that \(x_0\) is an approximate solution associated to a parameter \(\tau\), in Theorem 2.1 a solvability result for the inequality system is proved and a sharp upper bound of the ratio between the distance from \(x_0\) to the solution set and the distance from \(F(x_0)\) to the cone \(K\) is provided explicitly in terms of \(\tau\) and \(x_0.\) Theorems 2.2 and 2.3 deal with the particular case \(Y=\mathbb R^l\), where the conic inequality system is an inequality/equality system; here, the real-valued functions involved are supposed to be analytic. In particular, Theorem 2.2 improves the corresponding result due to \textit{J.-P. Dedieu} [SIAM J. Optim. 11, No. 2, 411--425 (2000; Zbl 1004.47042)].