## Nonsmooth Vector Functions and Continuous OptimizationA recent significant innovation in mathematical sciences has been the progressive use of nonsmooth calculus, an extension of the differential calculus, as a key tool of modern analysis in many areas of mathematics, operations research, and engineering. Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems and variational inequalities in finite dimensions. The treatment is motivated by a desire to expose an elementary approach to nonsmooth calculus by using a set of matrices to replace the nonexistent Jacobian matrix of a continuous vector function. Such a set of matrices forms a new generalized Jacobian, called pseudo-Jacobian. A direct extension of the classical derivative that follows simple rules of calculus, the pseudo-Jacobian provides an axiomatic approach to nonsmooth calculus, a flexible tool for handling nonsmooth continuous optimization problems. Illustrated by numerous examples of known generalized derivatives, the work may serve as a valuable reference for graduate students, researchers, and applied mathematicians who wish to use nonsmooth techniques and continuous optimization to model and solve problems in mathematical programming, operations research, and engineering. Readers require only a modest background in undergraduate mathematical analysis to follow the material with minimal effort. |

### What people are saying - Write a review

### Contents

1 | |

12 PseudoJacobian Matrices | 10 |

13 Nonsmooth Derivatives | 14 |

14 PseudoDifferentials and PseudoHessians of Scalar Functions | 23 |

15 Recession Matrices and Partial PseudoJacobians | 35 |

16 Constructing Stable PseudoJacobians | 40 |

17 Gâteaux and Frechet PseudoJacobians | 49 |

Calculus Rules for PseudoJacobians | 57 |

34 Convex Interior Mapping Theorems | 118 |

35 Metric Regularity and PseudoLipschitzian Property | 128 |

Nonsmooth Mathematical Programming Problems | 143 |

42 SecondOrder Conditions | 155 |

43 Composite Programming | 168 |

44 Multiobjective Programming | 186 |

Monotone Operators and Nonsmooth Variational Inequalities | 207 |

52 Generalized Convex Functions | 222 |

22 The Mean Value Theorem and Taylors Expansions | 66 |

23 A General Chain Rule | 82 |

24 Chain Rules Using Recession PseudoJacobian Matrices | 85 |

25 Chain Rules for Gateaux and Frechet PseudoJacobians | 93 |

Openness of Continuous Vector Functions | 98 |

32 Open Mapping Theorems | 110 |

33 Inverse and Implicit Function Theorems | 115 |

53 Variational Inequalities | 230 |

54 Complementarity Problems | 243 |

Bibliographical Notes | 255 |

259 | |

Notations | 265 |

266 | |

### Other editions - View all

Nonsmooth Vector Functions and Continuous Optimization V. Jeyakumar,Dinh The Luc No preview available - 2010 |

Nonsmooth Vector Functions and Continuous Optimization V. Jeyakumar,Dinh The Luc No preview available - 2007 |