The concave utilities in the basic network utility maximisation (NUM) problem are only suitable for elastic flows. In networks with both elastic and inelastic traffic, the utilities of inelastic traffic are usually modelled by the sigmoidal functions which are non-concave functions. Hence, the basic NUM problem becomes a non-convex optimisation problem. To address the non-convex NUM, the author...

The classical definition of Steiner symmetrizations functions are defined according to the function level sets and layered representation functions. In this paper, is not only transformed into one-dimensional parabolic functions, but also depends on log-concave To end we prove Steiner’s inequality layering in space

We consider the maximization of a gross substitutes utility function under budget constraints. This problem naturally arises in applications such as exchange economies in mathematical economics and combinatorial auctions in (algorithmic) game theory. We show that this problem admits a polynomial-time approximation scheme (PTAS). More generally, we present a PTAS for maximizing a discrete concav...

In this paper, the concepts of $L$-concave structures, concave $L$-interior operators and concave $L$-neighborhood systems are introduced. It is shown that the category of $L$-concave spaces and the category of concave $L$-interior spaces are isomorphic, and they are both isomorphic to the category of concave $L$-neighborhood systems whenever $L$ is a completely distributive lattice. Also, it i...

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 (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the S...

We consider the convex optimization problem P : minx{f(x) : x ∈ K} where f is convex continuously differentiable, and K ⊂ R is a compact convex set with representation {x ∈ R : gj(x) ≥ 0, j = 1, . . . ,m} for some continuously differentiable functions (gj). We discuss the case where the gj ’s are not all concave (in contrast with convex programming where they all are). In particular, even if th...

In this paper we show that for concave piecewise linear exchange economies every competitive equilibrium satisfies the property that the competitive allocation is a non-symmetric Nash bargaining solution with weights being the initial income of individual agents evaluated at the equilibrium price vector. We prove the existence of competitive equilibrium for concave piecewise linear exchange eco...

Consider a separable concave minimization problem with nondecreasing costs over a general ground set X ⊆ R+. We show how to efficiently approximate this problem to a factor of 1+2 in optimal cost by a single piecewise linear minimization problem over X. The number of pieces is linear in 1/2 and polynomial in the logarithm of certain ground set parameters; in particular, it is independent of the...

A class of probabilistic constrained programming problems are considered where the probabilistic constraint is of the form P{gi(x, ξ) ≥ 0, i = 1, . . . , r} ≥ p and the functions gi, i = 1, . . . , r are concave. It is shown that the x-function on the left hand side is logarithmic concave provided ξ has a logarithmic concave density. Special cases are mentioned and algorithmic solution of probl...