Applied mathematics

A method of construction of exhaustive family of upper convex approximations

I. M. Proudnikov

LG Electronics, Moscow

Abstract: It is shown how to construct Exhausters for a Lipschitz function $f$ at a point $x$ which is an important problem for optimization of such functions. At first the function $f$ is modified to some function $\tilde{f}$ and an exhaustive set of upper convex approximations is constructed for it whose subdifferentials at zero define the upper Exhauster of the function $\tilde{f}$ at the point $x$. A family $\Im$ of convex compact set pairs for the function $f$ is constructed. $\Im$ is called BiExhauster of the function $f$ at the point $x$. The exhaustive sets of upper and lower convex approximations of the function $f$ at the point $x$ are defined with the help of the set $\Im$. Convex compact sets from the upper Exhauster of the function $\tilde{f}$ are constructed as limit values of average integrals from gradients of the function $f$ along curves from a defined set of curves along which $\tilde{f}$ is almost everywhere differentiable. Bibliogr. 12. Il. 8.

Keywords: Lipschitz function, directional derivative, upper and lower convex approximations, upper and lower exhausters, BiExhauster, extremum points, extremum condition.

UDC: 517.977


Accepted: October 25, 2012