\(\gamma\)-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.


90C30 Nonlinear programming
49J52 Nonsmooth analysis
26B25 Convexity of real functions of several variables, generalizations
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