×

Generalized convexities of lower semicontinuous functions. (English) Zbl 0585.26008

Let \({\mathcal F}\) be a class of real-valued functions defined on a real interval I, and let \(L\subset I\) be also an interval. By using \({\mathcal F}\) the author defines three types of generalized convex functions as follows. A function \(g:L\to {\mathbb{R}}\) is said to be (1) \({\mathcal F}\)-convex on L if for all \(t',t''\in L,\) \(t'\neq t'',\) and each \(\lambda \in (0,1)\) there exists a function \(f\in {\mathcal F}\) satisfying the following conditions: \((C_ 1)\) f is continuous on \((t',t'');\) \((C_ 2)\quad f(t')=g(t'),f(t'')=g(t'');\) \((C_ 3)\quad g((1-\lambda)t'+\lambda t'')\leq f((1-\lambda)t'+\lambda t'');\) (2) \({\mathcal F}\)-convexlike if for all \(t',t''\in L,\) \(t'\neq t'',\) and at least one \(\lambda \in (0,1)\) there exists a function \(f\in {\mathcal F}\) satisfying the conditions \((C_ 1)\), \((C_ 2)\) and \((C_ 3)\); (3) uniformly \({\mathcal F}\)-convexlike if there is a \(\delta \in (0,1/2)\) such that for all \(t',t''\in L,\) \(t'\neq t'',\) and at least one \(\lambda \in [\delta,1-\delta]\) there exists a function \(f\in {\mathcal F}\) satisfying the conditions \((C_ 1)\), \((C_ 2)\) and \((C_ 3)\). The main result of the paper reveals conditions on \({\mathcal F}\) which assure that every \({\mathcal F}\)-convexlike (resp. uniformly \({\mathcal F}\)-convexlike) function \(g:L\to {\mathbb{R}}\) which is lower semicontinuous on the interior of L is \({\mathcal F}\)-convex on L.
Reviewer: W.W.Breckner

MSC:

26A51 Convexity of real functions in one variable, generalizations
26B25 Convexity of real functions of several variables, generalizations
90C25 Convex programming
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Behringer F.A., ZAMM 60 pp 201– (1980)
[2] Behringer F.A., ZAMM 60 pp T335– (1980)
[3] Behringer F.A., Math. Operationsforsch. u. Stat., Ser. Optimization 14 pp 163– (1983) · Zbl 0519.90065
[4] Hartwig, H. 1980-3.Generalizations of convexity by means of modified secants, 1–32. DM Karl Marx University of Economics Budapest.
[5] Hartwig H., Math. Operationsforsch u. Stat., Ser. Optimization 14 pp 49– (1983) · Zbl 0514.26003
[6] Schaible S., Generalized concavity in optimization and economics (1981)
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.