\(\gamma\)-subdifferential and \(\gamma\)-convexity of functions on the real line (Q2366092)

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.