## Multivalued Analysis and Nonlinear Programming Problems with PerturbationsThis book is concerned with topological and differential properties of multivalued mappings and marginal functions. Beside this applica tions to the sensitivity analysis of optimization problems, in particular nonlinear programming problems with perturbations, are studied. The elaborated methods are primarily obtained by theories and concepts of two former Soviet Union researchers, Demyanov and Rubinov. Con sequently, a significant part of the presented results have never been published in English before. Based on the use of directional derivatives as a key tool in studying nonsmooth functions and multifunctions, these results can be considered as a further development of quasidifferential calculus created by Demyanov and Rubinov. In contrast to other research in this field, especially the recent publica tion by Bonnans and Shapiro, this book analyses properties of marginal functions associated with optimization problems under quite general con straints defined by means of multivalued mappings. A unified approach to directional differentiability of functions and multifunctions forms the base of the volume. |

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### Contents

BASIC NOTATION | 1 |

BASIC CONCEPTS AND PROBLEMS OF MULTIVALUED ANALYSIS | 5 |

12 Convex Functions | 9 |

13 Topological and Differential Properties of Convex Functions | 12 |

2 Elements of Nonsmooth Analysis and Optimality Conditions | 14 |

22 Directional Derivatives | 15 |

23 Clarke Subdifferentials | 16 |

3 Quasidifferentiable Functions and Problems | 18 |

2 Weakly Uniformly Differentiable Multivalued Mappings | 84 |

3 Strongly Differentiable Mappings and Directional Differentiability of Marginal Functions | 90 |

32 Directional differentiability of marginal functions | 93 |

SENSITIVITY ANALYSIS | 97 |

1 Stability Properties of Optimal Solutions in Mathematical Programming Problems | 98 |

2 Regular Multivalued Mappings | 102 |

21 Regularity Conditions | 103 |

22 Rregular Mappings | 106 |

32 Necessary Optimality Conditions | 22 |

TOPOLOGICAL AND DIFFERENTIAL PROPERTIES OF MULTIVALUED MAPPINGS | 27 |

12 Marginal Functions | 32 |

13 Pseudolipschitz and Pseudoholder Continuity of Multivalued Mappings | 37 |

14 Properties of Convex Mappings | 39 |

15 Closed convex processes | 41 |

2 Directional Differentiability of Multivalued Mappings | 43 |

22 Description of Derivatives of Multivalued Mappings in Terms of the Distance Function | 46 |

23 Firstorder Approximations of Multivalued Mappings | 50 |

24 Some Properties of Derivatives of Multivalued Mappings | 51 |

3 Lemma About the Removal of Constraints | 54 |

SUBDIFFERENTIALS OF MARGINAL FUNCTIONS | 59 |

12 Pseudolipschitz Continuity and Metrical Regularity | 68 |

2 Locally Convex Mappings | 71 |

22 Subdifferentials of Marginal Functions for Locally Convex Multivalued Mappings | 73 |

DIRECTIONAL DERIVATIVES OF MARGINAL FUNCTIONS | 77 |

23 The Linear Tangent Cone and Derivatives of Regular Multivalued Mappings | 114 |

24 Subdifferentials of Marginal Functions in Regular Problems | 117 |

25 Secondorder Derivatives of Mappings | 119 |

26 Directional Regularity | 120 |

3 FirstOrder Directional Derivatives of Optimal Value Functions and Sensitivity Analysis of Suboptimal Solutions | 128 |

31 General Case | 129 |

32 Directional Derivatives of Optimal Value Functions in Nonlinear Programming Problems | 137 |

33 Holder Behaviour of Optimal Solutions and Directional Differentiability of Optimal Value Functions in Rregular Problems | 148 |

34 Problems with Vertical Perturbations | 165 |

35 Quasidifferentiable Programming Problems | 169 |

4 SecondOrder Analysis of the Optimal Value Function and Differentiability of Optimal Solutions | 175 |

Bibliographical Comments | 187 |

References | 191 |

203 | |

### Other editions - View all

Multivalued Analysis and Nonlinear Programming Problems with Perturbations Bernd Luderer,Leonid Minchenko,T. Satsura No preview available - 2010 |

Multivalued Analysis and Nonlinear Programming Problems with Perturbations Bernd Luderer,Leonid Minchenko,T. Satsura No preview available - 2012 |

### Common terms and phrases

A(zo arbitrary assume that yk Clarke subdifferential compact set constraints Convex Analysis convex cone convex function convex set Corollary Definition df(x dF(z Dini derivative direction x directional derivative directionally differentiable DlF(zq;x due to Lemma E F(x exists F is R)-regular F(xo function f gr F hi(z hi(zk holds inclusion inequality inf inf Lagrange multipliers Let F Let the assumptions Let the function Let the multivalued liminf limsup locally Lipschitz continuous lower semicontinuous mapping F marginal functions metrically regular multivalued mapping neighbourhood non-empty Nonlinear Programming obtain optimal solutions optimal value function p(xk p(xo perturbations point xo point zq problem Px Programming Problems Proof pseudolipschitz continuous quasidifferentiable regularity condition respect second-order sequence ek set of optimal statements are equivalent support function tangent cone Theorem u.s.c. at xo u(xo uj(xo upper semicontinuous upper w.u.d. V(xo V(yo valid vector

### Popular passages

Page 194 - On an algorithm solving two-level programming problems with nonunique lower level solutions.