Topic
Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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03 Jun 1996
TL;DR: A theory of “convex analysis” is developed for functions defined on integer lattice points and it is shown that a function ω has the exchange property if and only if it can be extended to a concave function such that the maximizers of (\bar \omega+any linear function) form an integral base polytope.
Abstract: A theory of “convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz's exchange property. It includes linear functions on matroids, valuations on matroids, and separable concave functions on the integral base polytope. It is shown that a function ω has the exchange property if and only if it can be extended to a concave function \(\bar \omega\)such that the maximizers of (\(\bar \omega\)+any linear function) form an integral base polytope. A Fenchel-type min-max theorem and discrete separation theorems are given, which contain, e.g., Frank's discrete separation theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection.
66 citations
TL;DR: The standard extension is proved to be identical to another type of concave extension, defined as the lower envelope of a class of affine functions majorizing the given polynomial.
Abstract: A well-known linearization technique for nonlinear 0–1 maximization problems can be viewed as extending any polynomial in 0–1 variables to a concave function defined on [0, 1]
n
. Some properties of this “standard” concave extension are investigated. Polynomials for which the standard extension coincides with the concave envelope are characterized in terms of integrality of a certain polyhedron or balancedness of a certain matrix. The standard extension is proved to be identical to another type of concave extension, defined as the lower envelope of a class of affine functions majorizing the given polynomial.
65 citations
TL;DR: In this paper, the authors studied the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions and proved integral representation formulae for such a first variation, which suggest to define in a natural way the notion of area measure for a log-concave function.
65 citations
64 citations
TL;DR: An algorithm is proposed for globally minimizing a concave function over a compact convex set by combining typical branch-and-bound elements like partitioning, bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided.
Abstract: An algorithm is proposed for globally minimizing a concave function over a compact convex set. This algorithm combines typical branch-and-bound elements like partitioning, bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided. In each step, essentially only few linear programming problems have to be solved. Some preliminary computational results are reported.
64 citations