Convex analysis and optimization MA 629              Darinka Dentcheva

Fall 2009                                                                     darinka.dentcheva@stevens.edu

 

“Namely, because the shape of the whole universe is most perfect and, in fact, designed by the wisest creator, nothing in all of the world will occur in which no minimum or maximum rule is somehow shining forth.” Leonard Euler (1744)

 

Objective

 

The objective of this course is to introduce the students to some basic results of convex analysis and to the theory of optimization. We shall develop necessary and sufficient optimality conditions of first and second order for nonlinear and possibly non-smooth optimization models with and without constraints; understand the role of constraint qualifications; present duality theory, and some stability results. The material in this course builds the necessary foundation for numerical methods of optimization and stochastic optimization (stochastic programming). Examples of optimization models from probability, statistics, approximation theory, mathematical finance, and other practical situations will be presented as well. These models will be used along the theoretical considerations to illustrate the discussed notions and phenomena, and to illustrate the scope of applications.

 

Time and Place

 

Wednesdays, 6:00—8:30 pm. in Babbio 110.

 

Office hours

 

Peirce 302 on Mondays 2:00--4:00 pm. or by appointment.

Tel: (201) 216-8640

Fax: (201) 216-8321

 

Course Material

 

Nonlinear Optimization, Andrzej Ruszczynski, Princeton University Press, 2006, ISBN-13: 978-0-691-11915-1; ISBN-10: 0-691-11915-5. 

 

Graded work

 

Six homework assignments will be given to help you understand and apply the concepts and the theoretical results. In addition, one midterm project and a final project will be assigned. The solutions will be discussed in class.

 

 

Plan of lectures

 

Date

Topic

Sep 2

Examples of optimization models. Introduction to convex sets.

Sep 9

Convex cones , normal cones, and separation theorems

Sep 16

Extreme points and representation of convex sets

Sep 23

Convex functions and convexity criteria, continuity properties

Sep 30

Differentiability properties of convex functions

Oct 7

Conjugate functions and their subgradients. Fenchel duality

Oct 14

Unconstrained minima.

Oct 21

Tangent cones and metric regularity

Oct 28

Optimality conditions for smooth problems

Nov 4

Optimality conditions for convex non-smooth  problems

Nov 11

Second order optimality conditions and sensitivity

Nov 18

Saddle points, dual problems

Dec 2

Duality relations

Dec 9

Convex relaxations of non-convex problems and decomposition

 

Final Exam