Faculty of Mathematics and Computer Science

Non-smooth analysis |

Code |
Semes-ter |
Hours: C+S+L |
Credits |
Type |
Section |

Teaching Staff in Charge |

Prof. MURESAN Marian, Ph.D., mmarian@math.ubbcluj.ro |

Aims |

Training up and improving the students' skill to deal with optimization problems with nonsmooth data. |

Content |

1. Introduction
1.1. Exemples of optimization problems having non-smooth data 2.1. Penalties and constraints 2. Multifunctions 2.1. Convergence of sets 2.2. Measurability of multifunctions, selection theorems 3. Subgradients 3.1. Definitions and main properies 3.2. Connection with derivatives and subderivatives 3.2. Rules of calculus 3.3. Cones: Clarke, Bouligand, intermediar, Ioffe, Mordukhovich 4. Geometry of non-smooth analysis 4.1. Variational principles 4.2. Proximal normals and subgradients 4.2. Limiting normals and subgradients 4.3. Duality 5. Subdifferential calculus 5.1. Calculus rules 5.2. Lagrange multipliers and applications 6. Applications 6.1. The method of adjoint arc 6.2. Applications in economy, equilibrium points 6.3. Applications in engeneering |

References |

1. Choset, H., Nonsmooth analysis, convex analysis, and their applications to motion planning, Intern. J. Comp. Geo. Appl., vol. 9(1999), nr. 4-5, 447-467.
2. Clarke, F. H., Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. 3. Jofre, A., A second-welfare theorem in non-convex economies, Canad. Math. J., vol. 27(2000), 175-184. 4. Loewen, P. D., Optimal Control and Nonsmooth Analysis, AMS, Providence, 1993. 5. Loewen, P. D., Rockafellar, T. R., The adjoint arc in nonsmooth optimization, SIAM J. Control Optim., 325(1991), 39-72. 6. Khan, M. A., Ioffe's normal cone and the foundations of welfare economics: the infinite dimensional theory, J. Math. Anal. Appl., vol. 161(1991), 284-298. 7. Mordukhovich, B., The extremal principle and its applications to optimixation and economics, in Optimization and related topics (A. Rubinov and B. Glover, eds.) Applied Optimization, vol. 47, Kluwer, 343-369. 8. Muresan, M., Analiza neneteda, Risoprint, Cluj-Napoca, 2001. 9. Rockafellar, T. R., Wets, R. J.-B., Variational Analysis, Springer, New York, 1998. |

Assessment |

A review and an exam. |