eBook

Self-Regularity

Terlaky, Tamás, Roos, Cornelis, Peng, Jiming

2009,2002,2003

Overview

Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work.

• Publisher: Princeton University Press
• Subject: Algorithm / Analysis of algorithms / Analytic function / Associative property / Barrier function / Block matrix / Combinatorial optimization / Combinatorics / Complexity / Conic optimization / Continuous optimization / Control theory / Convex optimization / Delft University of Technology / Derivative / Duality gap / Eigenvalues and eigenvectors / Embedding / Equation / Estimation / Explanation / Feasible region / Filter design / Function (mathematics) / General Topics for Engineers / Implementation / Instance (computer science) / Interior-point methods / Invertible matrix / Iteration / Jacobian matrix and determinant / Jordan algebra / Karush–Kuhn–Tucker conditions / Linear complementarity problem / Linear programming / Lipschitz continuity / Local convergence / Loss function / Mathematical optimization / MATHEMATICS / MATHEMATICS / Applied / MATHEMATICS / Optimization / Matrix function / McMaster University / Multiplication operator / Newton's method / Nonlinear programming / Optimal control / Optimization problem / Parameter / Pattern recognition / Polynomial / Programming (Mathematics) / Quadratic function / Requirement / Result / Scientific notation / Self-concordant function / Sensitivity analysis / Simplex algorithm / Simultaneous equations / Singular value / Smoothness / Solution set / Solver / Special case / Theorem / Theory / Time complexity / Variable (mathematics) / Variational inequality / Variational principle / Without loss of generality / Yurii Nesterov
• ISBN: 9780691091921, 0691091927, 0691091935, 9780691091938, 9781400825134, 140082513X