The concave utility in the Network Utility Maximization (NUM) problem is only suitable for elastic flows. However, the networks with the multiclass traffic, the utility of inelastic traffic is usually represented by the sigmoidal function which is a nonconcave function. Hence, the basic NUM problem becomes a nonconvex optimization problem. Solving the nonconvex NUM distributively is a difficult...

Nonconvex nonsmooth potential functions have superior restoration performance for the images with neat boundaries. However, several difficulties emerge from the numerical computation. Thus the graduated nonconvex (GNC) method is suggested to deal with these problems. To improve the performance of the GNC method further, a class of nonconvex nonsmooth approximate potential functions have been co...

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmoot...

This paper presents a multi-scale model in phase transitions of solid materials with both macro and micro effects. This model is governed by a semi-linear nonconvex partial differential equation which can be converted into a coupled quadratic mixed variational problem by the canonical dual transformation method. The extremality conditions of this variational problem are controlled by a triality...

Based on a study of a minimization problem, we present the following results applicable to possibly nonconvex sets in a Banach space: an approximate projection result, an extended extremal principle, a nonconvex separation theorem, a generalized Bishop-Phelps theorem and a separable point result. The classical result of Dieudonné (on separation of two convex sets in a finite dimensional space) ...

In this work we propose an outcome space approach for globally solving generalized concave multiplicative problems, a special class of nonconvex problems which involves the maximization of a finite sum of products of concave functions over a nonempty compact convex set. It is shown that this nonconvex maximization problem can be reformulated as an indefinite quadratic problem with infinitely ma...

Nonlinear convex optimization has provided both an insightful modeling language and a powerful solution tool to the analysis and design of communication systems over the last decade. A main challenge today is on nonconvex problems in these applications. This chapter presents an overview on some of the important nonconvex optimization problems in communication networks. Four typical applications...