**George****
****Mason**** ****University**

**Department
of SEOR and Mathematical Sciences Department**.

# Fall 2004

*Professor Roman A. Polyak*

Math 689/ IT 884-001: Advance Nonlinear Optimization

Tuesday 4:30-7:10 pm.
ENT 276

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**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.

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## 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.Courants penalty method for equality constrained optimization and its dual
equivalent-N.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.

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### Final Exam: December 14, 2004