Liqun Qi – författare

A collection of 28 refereed papers grouped according to four broad topics: duality and optimality conditions, optimization algorithms, optimal control, and variational inequality and equilibrium problems. Suitable for researchers, practitioners and postgrads.

Since its introduction by Isaac Newton (1669) and Joseph Raphson (1690) more than three hundred years ago, Newton's method or the Newton-Raphson method has become the most important technique for solving the system of smooth algebraic equations. Despite its simple structure, Newton's method possesses a fast local convergence rate - superlinear or quadratic. This outstanding feature of Newton's method leads to numerous extensions in the literature. Most of these extensions focus on systems of smooth equations. Since the 1980s, researchers the fields of optimization and numerical analysis have been working on extending Newton's method to non-differentiable system of algebraic equations. This book presents a comprehensive treatment of the development of the generalized Newton method for solving nonsmooth equations and related problems which grow out of science, engineering, economics and business and sheds light on further investigations of this fascinating topic oriented towards applications in optimization. Semismooth analysis, which form the backbone of further developments, is developed in Chapter 1. Topics then unfold systematically, with apposite illustrations and examples. Graduate students and researchers in this area will find the book useful.

This volume is a collection of papers that cover such diverse areas as linear and non-linear complementarity problems, variational inequality problems, non-smooth equations and non-smooth optimization problems, economic and network equilibrium problems, semi-definite programming problems, maximal monotone operator problems, and mathematical programs with equilibrium constraints. It provides useful tools for advancing the study of mathematical programming from both theoretical and practical aspects. The book is intended for students and researchers in optimization, mathematical programming, and operations research.

A collection of 28 refereed papers grouped according to four broad topics: duality and optimality conditions, optimization algorithms, optimal control, and variational inequality and equilibrium problems. Suitable for researchers, practitioners and postgrads.

The concept of 'reformulation' has long played an important role in mathematical programming. A classical example is the penalization technique in constrained optimization. More recent trends consist of reformulation of various mathematical programming problems, including variational inequalities and complementarity problems, into equivalent systems of possibly nonsmooth, piecewise smooth or semismooth nonlinear equations, or equivalent unconstrained optimization problems that are usually differentiable, but in general not twice differentiable. The book is a collection of peer-reviewed papers that cover such diverse areas as linear and nonlinear complementarity problems, variational inequality problems, nonsmooth equations and nonsmooth optimization problems, economic and network equilibrium problems, semidefinite programming problems, maximal monotone operator problems, and mathematical programs with equilibrium constraints. The reader will be convinced that the concept of 'reformulation' provides extremely useful tools for advancing the study of mathematical programming from both theoretical and practical aspects.Audience: This book is intended for students and researchers in optimization, mathematical programming, and operations research.

Tensors, or hypermatrices, are multi-arrays with more than two indices. In the last decade or so, many concepts and results in matrix theory – some of which are nontrivial – have been extended to tensors and have a wide range of applications (for example, spectral hypergraph theory, higher order Markov chains, polynomial optimization, magnetic resonance imaging, automatic control, and quantum entanglement problems). The authors provide a comprehensive discussion of this new theory of tensors.Tensor Analysis is unique in that it is the first book on the spectral theory of tensors; the theory of special tensors, including nonnegative tensors, positive semidefinite tensors, completely positive tensors, and copositive tensors; and the spectral hypergraph theory via tensors, which is covered in a chapter.

It also addresses higher-order diffusion tensor imaging, third-order symmetric and traceless tensors in liquid crystals, piezoelectric tensors, strong ellipticity for elasticity tensors, and higher-order tensors in quantum physics.