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Comp Math 310, Part B: Optimization

# Stat 310: Computational Math and Optimization II: Optimization; Part 1.

## Lectures.

• Lecture 1:
• 1.1: Course logistics.
• 1.2: Context of Optimization; Example Problems.
• 1.3: Modeling environments; AMPL. The best reference for AMPL is the AMPL web site.
• 1.4 Course Objectives.
• Lecture 2:
• 2.1 Newton’s method and implications.
• 2.2 Computing Derivatives.
• 2.3 Optimization Code Encapsulation.
• Lecture 3:
• 2.4 Linear Algebra.
• 2.5 Sparse Linear Algebra
• 3.1 Failure of Newton Method
• 3.2 Line Search Methods: Principles
• Lecture 4 (tentative plan)
• 3.2.2. Line Search Methods: Other considerations.
• 3.3 Dealing with Indefinite Matrices
• 3.4 Quasi Newton Methods
• Lecture 5
• Lecture 6
• Lecture 7
• 6.1 Matrix-free convergence framework
• 6.2 Krylov Methods in Optimization
• 6.3 Lanczos Algorithms
• 7.1 Nonlinear Least Squaares
• 7.2 Nonlinear Equations.
• Section 8 (from here on I refer to my own sectioning)
• 8.1 Dealing with nonsmoothness in NLP
• 8.2 Examples of NLP
• 8.3 The implicit Function Theorem
• 8.4 Optimiality Conditions for Equality-Constrained Optimization
• 8.5 Optimality Conditions for Inequality-Constrained Optimization
• Section 9
• 9.2 Augmented Lagrangian Approaches.
• Section 10
• 10.1 Types of Constrained Optimization Algorithms.
• 10.2 Merit Functions and Filters
• 10.3 Maratos Effect and Curvilinear Search
• Section 11.
• 11.1 Interior-point algorithms for quadratic programming.
• 11.2 Interior-point algorithms for general nonlinear programming.

## Homeworks

1. Homwework #1 Assigned as part of Prof Lim's assignment. Some useful files:
• The INTVAL package which contains automatic differentiation.
• If you install it, you can use my implementation of fenton's function. The files are: fenton.m and fenton_wrap.m ; the latter returns the function, gradient, and Hessian of the Fenton function. The headers of the functions should be fairly self-explanatory.
2. Homework #2, assigned 02/21/12 due 02/28/12
3. Homework #3, assigned 02/28/12 due 03/05/12
4. Homework #4, assgined 03/06/12 due 03/13/12

## Resources:

1. Textbook: Nocedal and Wright, "Numerical Optimization", Springer, 2006. Available online for free for members of the University of Chicago Community.
2. Excellent course page for a computation class from Maria Emelianenko from GMU.
3. Harvey Greenberg's Myths and Counterexamples in Mathematical Programming page on the INFORMS site.
4. The Matlab Exchange: A large set of free Matlab Programs contributed by Matlab Users. I start here many times when designing and developing a new project
6. Tim Kelley's collection of Matlab Programs for Optimization..
7. LSTRS, a Matlab large-scale trust-region algorithm implementation.
8. The AMPL modeling language web site.

## What I assume is known by the students.

1. Multivariate calculus.
2. First-order necessary optimality conditions for unconstrained optimization.
3. Linear algebra. Positive definite and semidefinite symmetric matrices, eigenvalues.
4. Beginner Matlab.