Optimality conditions for non-finite valued convex composite functions
Let \(F: \mathbb{R}^ n\to\mathbb{R}^ m\) be a locally Lipschitz function and let \(h: \mathbb{R}^ m\to\mathbb{R}\cup\{+\infty\}\) be a lower semicontinuous function. The paper concerns second order optimality conditions for the local minimizers of the composite function \(f=h\circ F\). The keystone of the authors' development is that a local minimizer of \(f\) is also a local minimizer of \(f_ \alpha=h_ \alpha\circ F\) for all sufficiently small \(\alpha>0\), where \(h_ \alpha(y)=\inf\{h(z)+\| y-z\|/\alpha\;:\;z\in \mathbb{R}^ m\}\). The results of the first author [Math. Program. 38, 287-302 (1987; Zbl 0641.49013)], where \(h\) is finite valued, are applied on the \(h_ \alpha\) case and the conclusion is derived taking the limit as \(\alpha\to 0\). A comparison is made with the results of \textit{R. T. Rockafellar} [Math. Oper. Res. 14, No. 3, 462-484 (1989; Zbl 0698.90070)] for the piecewise linear-quadratic case.
scientific article; zbMATH DE number 431045