Hiriart-Urruty, J.-B. From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality. (English) Zbl 0735.90056 Nonsmooth optimization and related topics, Proc. 4th Course Int. Sch. Math., Erice/Italy 1988, Ettore Majorana Int. Sci. Ser., Phys. Sci. 43, 219-239 (1989). [For the entire collection see Zbl 0719.00020.]The author presents a new necessary and sufficient condition for global optimality in nonconvex optimization. The problem addressed is that of minimizing globally a difference \(f=g-h\) of convex functions \(g\) and \(h\), or, which is an equivalent problem, that of maximizing globally a convex function \(h\) on a closed convex set \(C\). The necessary and sufficient conditions for global optimality involve \(\epsilon\)-subdifferentials of the component functions \(g\) and \(h\) (resp. the \(\epsilon\)-subdifferential of \(h\) and the set of \(\epsilon\)-normal directions to the set \(C\)). The conditions truly filter the local minima of \(f\) by the use of the parameter \(\epsilon \geq 0\). Reviewer: J.-B.Hiriart-Urruty Cited in 3 ReviewsCited in 75 Documents MSC: 90C26 Nonconvex programming, global optimization 49J52 Nonsmooth analysis 49K27 Optimality conditions for problems in abstract spaces 90C25 Convex programming 90C48 Programming in abstract spaces 58E30 Variational principles in infinite-dimensional spaces Keywords:necessary and sufficient condition; global optimality; nonconvex optimization; \(\epsilon\)-subdifferentials; dc-programming Citations:Zbl 0719.00020 PDFBibTeX XMLCite \textit{J. B. Hiriart-Urruty}, in: Compactness and boundedness for a class of concave-convex functions. . 219--239 (1989; Zbl 0735.90056)