Newton-Type Methods for Optimization and Variational Problems

1. Front Matter

   Pages i-xix

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2. Elements of Optimization Theory and Variational Analysis

   • Alexey F. Izmailov, Mikhail V. Solodov

   Pages 1-60

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3. Equations and Unconstrained Optimization

   • Alexey F. Izmailov, Mikhail V. Solodov

   Pages 61-137

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4. Variational Problems: Local Methods

   • Alexey F. Izmailov, Mikhail V. Solodov

   Pages 139-203

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5. Constrained Optimization: Local Methods

   • Alexey F. Izmailov, Mikhail V. Solodov

   Pages 205-303

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6. Variational Problems: Globalization of Convergence

   • Alexey F. Izmailov, Mikhail V. Solodov

   Pages 305-366

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7. Constrained Optimization: Globalization of Convergence

   • Alexey F. Izmailov, Mikhail V. Solodov

   Pages 367-437

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8. Degenerate Problems with Nonisolated Solutions

   • Alexey F. Izmailov, Mikhail V. Solodov

   Pages 439-541

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9. Back Matter

   Pages 543-573

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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.

• Complementarity problems
• Newton method
• Optimization
• Sequential quadratic programming
• Variational problems

“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)

• Moscow State University, Moscow, Russia

  Alexey F. Izmailov

• Instituto de Matem´atica Pura e Aplicada, Rio de Janeiro, Brazil

  Mikhail V. Solodov

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