Approximate Solutions for Abstract Inequality Systems
Chong Li, K. F. Ng · SIAM Journal on Optimization · 2013
We consider conic inequality systems of the type $F(x)\ge_K 0$, with approximate solution $x_0$ associated to a parameter $\tau$, where $F$ is a twice Fréchet differentiable function between Hilbert spaces $X$ and $Y$, and $\ge_K$ is the partial order in $Y$ defined by a nonempty convex (not necessarily closed) cone $K\subseteq Y$. We prove that, under the suitable conditions, the system $F(x)\ge_K 0$ is solvable, and the ratio of the distance from $x_0$ to the solution set $S$ over the distance from $F(x_0)$ to the cone $K$ has an upper bound given explicitly in terms of $\tau$ and $x_0$. 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.