Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

Power grids, flexible manufacturing, cellular communications: interconnectedness has consequences. This remarkable book gives the tools and philosophy you need to build network models detailed enough to capture essential dynamics but simple enough to expose the structure of effective control solutions and to clarify analysis. Core chapters assume only exposure to stochastic processes and linear algebra at the undergraduate level; later chapters are for advanced graduate students and researchers/practitioners. This gradual development bridges classical theory with the state-of-the-art. The workload model that is the basis of traditional analysis of the single queue becomes a foundation for workload relaxations used in the treatment of complex networks. Lyapunov functions and dynamic programming equations lead to the celebrated MaxWeight policy along with many generalizations. Other topics include methods for synthesizing hedging and safety stocks, stability theory for networks, and techniques for accelerated simulation. Examples and figures throughout make ideas concrete. Solutions to end-of-chapter exercises available on a companion website.

http://www.meyn.ece.ufl.edu/archive/spm_files/Presentations/Atlanta05.pdf

Control Techniques for Complex Networks

We develop a programmatic procedure for establishing the stability of queueing networks and scheduling policies. The method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a Lyapunov function. If the underlying system is Markovian, our method establishes not only positive recurrence and the existence of a steady-state probability distribution, but also the geometric convergence of an exponential moment. We illustrate this method on several example problems. >

Stability of queueing networks and scheduling policies

A procedure based on Lemke's algorithm is developed which either computes stationary points for general quadratic programs or else shows that the program has no optimum. If a general quadratic program has an optimum and satisfies a non-degeneracy condition then it is demonstrated that there are an odd number of stationary points.

On Quadratic Programming

Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.

/pdf/copositive-programming-a-survey-il08v03kcv.pdf

Copositive Programming – a Survey

On classes of copositive matrices

: A differentiable function on a convex set K is said to be K-flat if its gradient vanishes at each of its zeros belonging to K. K-flatness of quadratic forms implies their constrained semi-definiteness. This furnishes a useful characterization of positive semi-definite matrices (when K is the whole space) and 'copositive-plus' matrices (when K is the non-negative orthant). Parallel comparisions between these classes of matrices are made. (Author)

Quadratic forms semi-definite over convex cones.

L. HALLER and I. G. T. MILLER: Direct Hypercone Unconstrained Minimization G. HORNE and G. S. TRACZ: Nonlinear Programming and Second Variation Schemes in Constrained Optimal Control Problems G. ZOUTENDIJK: On Continuous Finite-Dimensional Constrained Optimization PART VIII. PIVOTAL METHODS

G. J. Habetler

Papers

On classes of copositive matrices

Quadratic forms semi-definite over convex cones.

L. HALLER and I. G. T. MILLER: Direct Hypercone Unconstrained Minimization G. HORNE and G. S. TRACZ: Nonlinear Programming and Second Variation Schemes in Constrained Optimal Control Problems G. ZOUTENDIJK: On Continuous Finite-Dimensional Constrained Optimization PART VIII. PIVOTAL METHODS