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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 \(\epsilon \geq 0\).

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

Citations:

Zbl 0719.00020
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