Department of SEOR and Mathematical Sciences Department.
Professor Roman A. Polyak
Math 689/ IT 884-001: Advance Nonlinear Optimization
Tuesday ENT 276
Office: Room127, ST-II building; phone: 703-9931685; fax: 703-9931521
Office Hours: Thursday or by appointment. E-mail: firstname.lastname@example.org
Text: D.Bertsekas “Nonlinear
Programming Second Edition Athena Scientific,
S.Nash,A.Sofer “Linear and Nonlinear Programming” The McGraw-Hill Companies Inc.1996.
Course Summary: A number of real life applications
arising in statistical learning theory, structural optimization, antennae design,
optimal power flow, radiation therapy planning, signal processing, economics
and finance just to mention a few lead to Nonlinear Programming (NLP).A general
NLP problem consists of finding a minimum (maximum) of a nonlinear function
under nonlinear constraints both inequalities and equations. In this course
along with classical NLP chapters that go back to
In the first part of the course we will consider theory and methods for unconstrained optimization as well as NLP with equality constraints. We will also cover elements of convex analysis and convex optimization theory including optimality criteria and convex duality.
In the second part of the course we will cover recent advances in NLP including Interior Point Methods(IPMs) and Nonlinear Rescaling (NR) theory and methods in constrained optimization. Particular emphasis will be given to the primal-dual approaches based on IPM and NR.
There will be homework assignment and projects.
Grading: 15% homework; 30% midterm exam; 20 % project; 35 % final exam.
8. Sequential unconstrained minimization technique (SUMT).Classical barrier and distance functions .
9. Interior Point Method for NLP.
10. Augmented Lagrangian. Lagrange multipliers method for equality constraints and its dual equivalent-quadratic prox-method for unconstrained optimization.
11. Nonlinear Rescaling (NR) principle for inequality constraint optimization. Modified barrier functions, modified distance functions and correspondent methods.
12. NR multipliers methods and their dual equivalent-interior prox with entropy-like distance functions.
13. Primal-dual Interior Point Methods.
14. Primal-dual NR methods in constrained optimization. Numerical realization and numerical results.