Strong KKT conditions and weak sharp solutions in convex-composite optimization (Q623359)

Let \(X\) be a Banach space, \(\phi _{0},...,\phi _{m}:X\longrightarrow \mathbb{R}\cup \{+\infty \}\), and \(C\subseteq X\) be a closed convex set. The authors consider the problem of minimizing \(\phi_{0}(x)\) subject to \(\phi _{i}(x)\leq 0\) (\(i=1,...,m\)) and \(x\in C\), under the assumption that, for each \(i,\) one has \(\phi _{i}=\psi _{i}\circ f_{i}\) for a l.s.c. proper convex function \(\psi _{i}\) on a Banach space \(Y\) and a smooth function \(f_{i}:X\longrightarrow Y.\) They define several strong KKT conditions and show them to be related to the notions of sharp and weak sharp minima (in the sense of \textit{J. V. Burke} and \textit{M. C. Ferris} [SIAM J. Control Optimization 31, No. 5, 1340--1359 (1993; Zbl 0791.90040)]) of the penalty function \(p_{\tau }(x):=\phi _{0}(x)+\tau \sum_{i=1}^{m}\max\{\phi _{i}(x),0\}+\tau d(x,C)\). The results are applied to the study of metric regularity for convex-composite inequalities.