×

\(\gamma\)-subdifferential and \(\gamma\)-convexity of functions on a normed space. (English) Zbl 0831.90105

Summary: The \(\gamma\)-subdifferential \(\partial_\gamma\) is introduced for investigating the global behavior of real-valued functions on a normed space \(X\). If \(f: D\subset X\to \mathbb{R}\) attains its global minimum on \(D\) at \(x^*\), then \(0\in \partial_\gamma f(x^*)\). This necessary condition always holds, even if \(f\) is not continuous or \(x^*\) is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable \(\gamma\in \mathbb{R}_+\), many local minima cannot satisfy this necessary condition.
For the sufficient conditions, the so-called \(\gamma\)-convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a \(\gamma\)-convex function. There are \(\gamma\)-convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two \(\gamma\)-convex functions. For all that, \(\gamma\)-convex functions still have properties similar to those of convex functions. For instance, each \(\gamma\)-local minimizer of \(f\) is at the same time a global one. If \(f\) attains its global minimum on \(D\), then it does so at least at one point of its \(\gamma\)-boundary.

MSC:

90C30 Nonlinear programming
49J52 Nonsmooth analysis
26B25 Convexity of real functions of several variables, generalizations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Horst, R., andTuy, H. Global Optimization: Deterministic Approaches, Springer Verlag, New York, New York, 1990. · Zbl 0704.90057
[2] Hartwig, H.,On Generalized Convex Functions, Optimization, Vol. 14, pp. 49–60, 1983. · Zbl 0514.26003
[3] Hu, T. C., Klee, V., andLarman, D.,Optimization of Globally Convex Functions, SIAM Journal on Control and Optimization, Vol. 27, pp. 1026–1047, 1989. · Zbl 0686.52006
[4] Hiriart-Urruty, J. B.,From Convex Optimization to Nonconvex Optimization, Part 1: Necessary and Sufficient Conditions for Global Optimality, Nonsmooth Optimization and Related Topics, Edited by F. H. Clarke, V. F. Demyanov, and F. Giannessi, Plenum Press, New York, New York, 1989. · Zbl 0735.90056
[5] Clarke, F. H.,Optimization and Nonsmooth Analysis, Les Publications CRM, Montréal, Québec, Canada, 1989. · Zbl 0727.90045
[6] Magasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.
[7] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972. · Zbl 0224.49003
[8] Roberts, A. W., andVarberg, D. E.,Convex Functions, Academic Press, New York, New York, 1973. · Zbl 0271.26009
[9] Natanson, I. P.,Theorie der Funktionen einer Reellen Veränderlichen, Akademie Verlag, Berlin, Germany, 1969. · Zbl 0056.05201
[10] Phú, H. X.,{\(\gamma\)}-Subdifferential and {\(\gamma\)}-Convexity of Functions on the Real Line, Applied Mathematics and Optimization, Vol. 27, pp. 145–160, 1993. · Zbl 0798.49024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.