This second edition of the book includes a thorough theoretical and computational analysis of unconstrained and constrained optimization algorithms. The qualifier modern in the title refers to the unconstrained and constrained optimization algorithms that combine and integrate the latest and the most efficient optimization techniques and advanced computational linear algebra methods. A prime concern of this book is to understand the nature, purposes and limitations of modern nonlinear optimization algorithms. This clear, friendly and rigorous exposition discusses in an axiomatic manner the theory behind the nonlinear optimization algorithms for understanding their properties and their convergence. The presentation of the computational performances of the most known modern nonlinear optimization algorithms is a priority.

The book is designed for self-study by professionals or undergraduate or graduate students with a minimal background in mathematics, including linear algebra, calculus, topology and convexity. It is addressed to all those interested in developing and using new advanced techniques for solving large-scale unconstrained or constrained complex optimization problems. Mathematical programming researchers, theoreticians and practitioners in operations research, practitioners in engineering and industry researchers, as well as graduate students in mathematics, Ph.D. and master in mathematical programming will find plenty of recent information and practical approaches for solving real large-scale optimization problems and applications.

Provides theoretical background which gives insights into how the methods are derived Presents new theoretical aspects of the unconstrained and constrained optimization in an axiomatic manner Includes an Appendix covering mathematical concepts from computational linear algebra, calculus, topology, and convexity

Autorentext

Neculai Andrei holds a position of research professor at the Center for Advanced Modeling and Optimization at the Academy of Romanian Scientists in Bucharest, Romania. Dr. Andrei’s areas of interest include mathematical modeling, linear programming, nonlinear optimization, high performance computing and numerical methods in mathematical programming. His articles and books give careful attention to both mathematical rigor and practical relevance. In all his publications he focuses on understanding the nature, purpose and limitations of nonlinear optimization algorithms and applied mathematics in general. In addition to this present volume, Neculai Andrei published several books with Springer: Modern Numerical Nonlinear Optimization, first edition (2022), A Derivative-free Two-Level Random Search Method for Unconstrained Optimization (2021), Nonlinear Conjugate Gradient Methods for Unconstrained Optimization (2020), Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology (2017), and Nonlinear Optimization Applications Using the GAMS Technology (2013).

Inhalt

1. Introduction.- 2. Fundamentals on unconstrained optimization.-3 . Steepest descent method.- 4. Newton method.- 5. Conjugate gradient methods.- 6. Quasi-Newton methods.- 7. Inexact Newton method.- 8. Trust-region method.- 9. Direct methods for unconstrained optimization.- 10. Optimality conditions for nonlinear optimization.- 11. Constrained nonlinear optimization methods.- 12. Simple bound optimization.- 13. Quadratic programming.- 14. Penalty and augmented Lagrangian.- 15. Sequential quadratic programming.- 16. Primal methods. The generalized reduced gradient with sequential linearization. -17. Interior-point methods.- 18. Filter methods.- 19. Interior-point filter line search (IPOPT).- 20. Direct methods for constrained optimization.- Appendix A. Mathematical review.- Appendix B. SMUNO collection. Small scale optimization applications.- Appendix C. LACOP collection. Large-scale continuous nonlinear optimization applications.- Appendix D. MINPACK-2 collection. Large-scale unconstrained optimization applications.- References.- Author Index.- Subject Index.