George Mason University – Graduate Council
Graduate Course Approval Form
All courses numbered 500 or above must be submitted to the Graduate Council for final approval after approval by the
sponsoring College, School or Institute.
Graduate Council requires submission of this form for a new course or any change to existing courses. For a new course,
please attach a copy of the syllabus and catalog description (with catalog credit format, e.g. 3:2:1). The designated
representative of the College, School or Institute should forward the form along with the syllabus and catalog description, if
required, as an email attachment (in one file) to the secretary of the Graduate Council. A printed copy of the form with
signatures and the attachments should be brought to the Graduate Council meeting. Please complete the Graduate Course
Coordinator Form if the proposed changes will affect other units.
Note: Colleges, Schools or Institutes are responsible for submitting new or modified catalog descriptions (35 words or
less, using catalog format) to Creative Services by deadlines outlined in the yearly Catalog production calendar.
Please indicate: New___X___
Modify______
Delete_______
Department/Unit:____ SEOR____________ Course Subject/Number:___ ___OR 784______________
Submitted by:________ C.-H. Chen____________ Ext:__ 3-3572 ___ Email:____ cchen9@gmu.edu______
Course Title:__________ Advanced Topics in Nonlinear Programming _________________________
Effective Term (New/Modified Courses only): _ Fall 2007 __
Credit Hours: (Fixed) _3___
(Var.) __3__ to __0__
Final Term (deleted courses only):____________
Grade Type (check one):
__X__
_____
_____
Regular graduate (A, B, C, etc.)
Satisfactory/No Credit only
Special graduate (A, B, C, etc. + IP)
Repeat Status*(check one): _X_ NR-Not repeatable ____ RD-Repeatable within degree ____ RT-Repeatable within term
*Note: Used only for special topics, independent study, or internships courses
Total Number of Hours Allowed: _______
Schedule Type Code(s): 1._X_ LEC=Lecture SEM=Seminar STU=Studio INT=Internship IND=Independent Study
2.____ LAB=Lab RCT=Recitation (second code used only for courses with Lab or Rct component)
Prereq __X_ Coreq ___ (Check one): OR 644
_______________________________________________________________
__________________________________________________________________________________________
Note: Modified courses - review prereq or coreq for necessary changes; Deleted courses - review other courses to correct prereqs that list the deleted course.
Description of Modification (for modified courses):_Crosslist with IT 784 __
Special Instructions (major/college/class code restrictions, if needed):__________________________________________
Department/Unit Approval Signature:_________________________________________ Date: _____________
College/School Committee Approval Signature:__________________________________ Date:_____________
Graduate Council Approval Date:____________ Provost Office Signature:_________________________________
George Mason University
Graduate Course Coordination Form
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George Mason University
Department of SEOR
Fall 2004
Professor Roman A. Polyak
OR 784-001: Advance Nonlinear Optimization
Tuesday 4:30-7:10 pm. ENT 276
Office: Room127, ST-II building; phone: 703-9931685; fax: 703-9931521
Office Hours: Thursday 3 pm-5 pm or by appointment. E-mail: rpolyak@gmu.edu
Text: D.Bertsekas “Nonlinear Programming Second Edition Athena Scientific,Belmont,Massachusetts.
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 Newton, Cauchy and Lagrange we
will cover recent advances and trends in NLP.
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.
Course Schedule
1. Real life applications and mathematical problems that lead to NLP formulation.
2. Basics in unconstrained optimization: gradient method, Newton method and their modifications.
3. Optimization problems with equality constraints. Lagrangian equations as necessary optimality
condition. Lagrangian duality: the dual functions and the dual problem.
4. R.Courant’s penalty method for equality constrained optimization and its dual equivalentN.Tichonov’s regularization method for unconstrained optimization.
5. Convex functions, convex sets and the convex optimization problem.
Karush-Kuhn-Tucker’s optimality condition. Elements of the duality theory in convex optimization.
6. Principle of feasible directions and first primal-dual method for convex optimization.
7 Midterm
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.
Final Exam: December 14, 2004
Catalog Description:
784 Advanced Topics in Nonlinear Programming (3:3:0)Prerequisite: OR 644. Studies theory and
algorithms for solving nonlinear optimization problems. Contents vary; possible topics include large-scale and
parallel-unconstrained optimization, theoretical issues in constrained optimization, duality theory, Lagrangian
and sequential quadratic programming methods. May be repeated for credit when topics are distinctly different.