Convexity and Optimization in Finite Dimensions I

Abstract

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Convexity and Steinitz's exchange property

“…agrees with the support function of B as defined in [36], [40]. Obviously, ψ • (p) is concave, ψ • (0) = 0, and positively homogeneous, i.e., ψ…”

“…In line with the standard method in convex analysis [36], [40], we introduce the concept of conjugate functions.…”

“…We call g • the concave conjugate function of g. Since B is finite, g • is a polyhedral concave function [36], [40], taking finite values for all p. Furthermore we definê g :…”

“…The theory of convex/concave functions (Rockafellar [36], Stoer-Witzgall [40]) has played the core role in the field of nonlinear optimization as well as in other fields of mathematical sciences. Convex/concave functions are computationally tractable by virtue of the following two facts:…”

Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients

“…The rate function I is the convex conjugate, or Fenchel-Legendre transform, of log(Π t ) (see [16]; [51], sect. 4.6; or [47], sect.…”