$$\gamma$$-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.

MSC:

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

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