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\(\gamma\)-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.

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
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References:

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