\(\gamma\)-subdifferential and \(\gamma\)-convexity of functions on the real line. (English) Zbl 0798.49024
For a function \(f: D\to\mathbb{R}\), \(D\) interval in \(\mathbb{R}\) and \(\gamma: \mathbb{R}\to \mathbb{R}_ +^*\), the \(\gamma\)-subdifferential of \(f\) at \(x\in D\) is defined as the set of all numbers \(c\in\mathbb{R}\) for which there exist \(x_ 1\), \(x_ 2\) such that \(x\in [x_ i, x_ i+ \gamma(x_ i)]\) for \(i\in \{1,2\}\) and
\[
{{f(x_ 1+ \gamma(x_ 1))- f(x_ 1)} \over {\gamma(x_ 1)}} \leq c\leq {{f(x_ 2+ \gamma(x_ 2))- f(x_ 2)} \over {\gamma(x_ 2)}}.
\]
The relations with the usual subdifferential are studied and a related \(\gamma\)-convexity is derived.
Necessary and/or sufficient conditions for global optimality are given in terms of \(\gamma\)-subdifferential.
Necessary and/or sufficient conditions for global optimality are given in terms of \(\gamma\)-subdifferential.
Reviewer: V.Anisiu (Cluj-Napoca)
MSC:
49K27 | Optimality conditions for problems in abstract spaces |
49K05 | Optimality conditions for free problems in one independent variable |
26A51 | Convexity of real functions in one variable, generalizations |
46G05 | Derivatives of functions in infinite-dimensional spaces |
52A01 | Axiomatic and generalized convexity |
65K05 | Numerical mathematical programming methods |
90C25 | Convex programming |
90C48 | Programming in abstract spaces |
Keywords:
\(\gamma\)-convexityReferences:
[1] | Clarke FH (1989) Optimization and Nonsmooth Analysis, Les Publications CRM, Montr?al · Zbl 0727.90045 |
[2] | Hartwig H (1983) On generalized convex functions. Optimization 14:49-60 · Zbl 0514.26003 · doi:10.1080/02331938308842832 |
[3] | Hiriart-Urruty JB (1989) From convex optimization to nonconvex optimization, Part I: Necessary and sufficient conditions for global optimality. In: Nonsmooth Optimization and Related Topics (Clarke FH, Demyanov VF, Giannessi F, eds). Plenum, New York, pp 219-239 · Zbl 0735.90056 |
[4] | Hu TC, Klee V, Larman D (1989) Optimization of globally convex functions. SIAM J Control Optim 27:1026-1047 · Zbl 0686.52006 · doi:10.1137/0327055 |
[5] | Roberts AW, Varberg DE (1973) Convex Functions. Academic Press, New York · Zbl 0271.26009 |
[6] | Rockafellar RT (1972) Convex Analysis. Princeton University Press, Princeton, NJ |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.