Nonlinear programming. Theory, algorithms, and applications.

*(English)*Zbl 0563.90068
A Wiley-Interscience Publication. New York: John Wiley & Sons, Inc. XVII, 444 p. (1983).

This is an introduction to nonlinear programming, suitable as a text for graduate students, or possibly advanced undergraduates. Each chapter is followed by a problem section. The development is quite rigorous, with proofs of most convergence results included. The author has carefully included realistic examples to illustrate the sorts of problems that can be solved by the algorithms.

A chapter by chapter summary follows: Chapter 1: The nature of optimization problems. This is a discussion of some of the practical problems that arise when one attempts to model real world problems. Chapter 2: Analytical background. This chapter introduces needed tools from convexity, matrix theory, and analysis, and discusses optimality conditions for unconstrained problems. Chapter 3: Factorable functions. This is a new approach developed by the author to represent nonlinear functions in a computationally oriented way. Chapter 4: Unconstrained optimization models. Examples are given to motivate the theoretical and algorithmic material in Chapters 5-9. Chapter 5: Minimizing a function of a single variable. It includes Newton’s method, secant, bisection, quadratic fit, and golden section. Chapter 6: General convergence theory for unconstrained minimization algorithms. Sufficient conditions are established guaranteeing that an accumulation point satisfy either first- or second-order optimality conditions. Chapter 7: Newton’s method with variations. Chapter 8: Conjugate direction algorithms. Chapter 9: Quasi- Newton algorithms. Chapter 10: First- and second-order optimality conditions (for constrained problems). Chapter 11: Applications of optimality conditions. Sensitivity analysis and rates of convergence are discussed here. Chapter 12: Models with linear constraints. Contains a discussion of some practical examples of linearly constrained problems with nonlinear objectives. Chapter 13: Variable-reduction algorithms. These algorithms for linearly constrained nonlinear programming problems are primal feasible, and each iteration can be viewed as an unconstrained minimization on a space of reduced dimension. Chapter 14: Models with nonlinear constraints. This chapter contains more examples. Chapter 15: Direct algorithms for nonlinearly constrained problems. These methods imitate the methods of Chapter 13. Chapter 16: Sequential unconstrained minimization techniques. Includes barrier and penalty function methods, as well as a short discussion of augmented Lagrangian techniques. Chapter 17: Sequential constraint linearization techniques. Includes methods that solve a sequence of linarly constrained problems. Chapter 18: Obtaining global solutions. Chapter 19: Geometric programming.

A chapter by chapter summary follows: Chapter 1: The nature of optimization problems. This is a discussion of some of the practical problems that arise when one attempts to model real world problems. Chapter 2: Analytical background. This chapter introduces needed tools from convexity, matrix theory, and analysis, and discusses optimality conditions for unconstrained problems. Chapter 3: Factorable functions. This is a new approach developed by the author to represent nonlinear functions in a computationally oriented way. Chapter 4: Unconstrained optimization models. Examples are given to motivate the theoretical and algorithmic material in Chapters 5-9. Chapter 5: Minimizing a function of a single variable. It includes Newton’s method, secant, bisection, quadratic fit, and golden section. Chapter 6: General convergence theory for unconstrained minimization algorithms. Sufficient conditions are established guaranteeing that an accumulation point satisfy either first- or second-order optimality conditions. Chapter 7: Newton’s method with variations. Chapter 8: Conjugate direction algorithms. Chapter 9: Quasi- Newton algorithms. Chapter 10: First- and second-order optimality conditions (for constrained problems). Chapter 11: Applications of optimality conditions. Sensitivity analysis and rates of convergence are discussed here. Chapter 12: Models with linear constraints. Contains a discussion of some practical examples of linearly constrained problems with nonlinear objectives. Chapter 13: Variable-reduction algorithms. These algorithms for linearly constrained nonlinear programming problems are primal feasible, and each iteration can be viewed as an unconstrained minimization on a space of reduced dimension. Chapter 14: Models with nonlinear constraints. This chapter contains more examples. Chapter 15: Direct algorithms for nonlinearly constrained problems. These methods imitate the methods of Chapter 13. Chapter 16: Sequential unconstrained minimization techniques. Includes barrier and penalty function methods, as well as a short discussion of augmented Lagrangian techniques. Chapter 17: Sequential constraint linearization techniques. Includes methods that solve a sequence of linarly constrained problems. Chapter 18: Obtaining global solutions. Chapter 19: Geometric programming.

##### MSC:

90Cxx | Mathematical programming |

49Mxx | Numerical methods in optimal control |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |