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This book presents comprehensive state-of-the-art theoretical analysis of the fundamental Newtonian and Newtonian-related approaches to solving optimization and variational problems. A central focus is the relationship between the basic Newton scheme for a given problem and algorithms that also enjoy fast local convergence. The authors develop general perturbed Newtonian frameworks that preserve fast convergence and consider specific algorithms as particular cases within those frameworks, i.e., as perturbations of the associated basic Newton iterations. This approach yields a set of tools for the unified treatment of various algorithms, including some not of the Newton type per se. Among the new subjects addressed is the class of degenerate problems. In particular, the phenomenon of attraction of Newton iterates to critical Lagrange multipliers and its consequences as well as stabilized Newton methods for variational problems and stabilized sequential quadratic programming for optimization. This volume will be useful to researchers and graduate students in the fields of optimization and variational analysis.
List of contents
1. Elements of optimization theory and variational analysis.- 2. Equations and unconstrained optimization.- 3. Variational problems: local methods.- 4. Constrained optimization: local methods.- 5. Variational problems: globalization of convergence.- 6. Constrained optimization: globalization of convergence.- 7. Degenerate problems with non-isolated solutions.- A. Miscellaneous material.
Summary
This book presents comprehensive state-of-the-art theoretical analysis of the fundamental Newtonian and Newtonian-related approaches to solving optimization and variational problems. A central focus is the relationship between the basic Newton scheme for a given problem and algorithms that also enjoy fast local convergence. The authors develop general perturbed Newtonian frameworks that preserve fast convergence and consider specific algorithms as particular cases within those frameworks, i.e., as perturbations of the associated basic Newton iterations. This approach yields a set of tools for the unified treatment of various algorithms, including some not of the Newton type per se. Among the new subjects addressed is the class of degenerate problems. In particular, the phenomenon of attraction of Newton iterates to critical Lagrange multipliers and its consequences as well as stabilized Newton methods for variational problems and stabilized sequential quadratic programming for optimization. This volume will be useful to researchers and graduate students in the fields of optimization and variational analysis.
Additional text
“This book covers Newton-type methods (in a broad sense) for the solution of optimization and variational problems (like variational inequalities and complementarity problems). … Several results presented in this book are new and based on recent publications, and cannot be found in any other monograph. … a useful reference for researchers and graduate students working in the field of optimization and variational analysis.” (Christian Kanzow, Mathematical Reviews, July, 2015)
Report
"This book covers Newton-type methods (in a broad sense) for the solution of optimization and variational problems (like variational inequalities and complementarity problems). ... Several results presented in this book are new and based on recent publications, and cannot be found in any other monograph. ... a useful reference for researchers and graduate students working in the field of optimization and variational analysis." (Christian Kanzow, Mathematical Reviews, July, 2015)
