Instructor: Laurent O. Jay, 225L MLH, ljay@math.uiowa.edu

Classes: at 10:30-11:20 MWF in 214 MLH

Prerequisites: 22M:100 or equivalent. A knowledge of computer programming.

22M:174/22C:174 deals with numerical optimization techniques. Optimization is becoming more important than ever before in the modern world of science and industry. Optimization algorithms are being called on to handle problems that are much larger and more mathematically complex than in years past. The best methods available today are complex, and their manner of operation is far from obvious.

Course description: This course will cover modern optimization techniques for both unconstrained and constrained optimization with continuous (as opposed to discrete) variables. Given its strong links to optimization techniques, the numerical solution of nonlinear equations will also be considered. At the end of the course the student should master most of the issues in numerical optimization.

My classnotes based on several sources will be available to students. See http://www.math.uiowa.edu/~ljay/m174_07.html for current information.

Contents: Characterization of solutions (such as optimality conditions in optimization) and convergence analysis of the algorithms will be essential to this course. We give below a partial list of topics and algorithms to be treated in connexion with three general classes of problems:

1. Unconstrained optimization: Steepest-descent method; Newton-like methods; Quasi-Newton methods; Linear/nonlinear conjugate gradient methods; Interval reduction methods; Line-search methods; Trust-region methods; Local and global convergence;
2. Nonlinear equations: Newton's method; Modified Newton's methods; Broyden's (quasi-Newton) method; Inexact Newton methods; The bisection method; Line-search methods and merit functions; Trust-region methods; Local and global convergence;
3. Constrained optimization with (in)equality constraints: Lagrange multipliers; Karush-Kuhn-Tucker conditions; Line-search methods and merit functions; Active-set methods (for inequality constraints); Penalty function methods (for equality constraints); Reduced-gradient and gradient-projection methods; Augmented Lagrangian and projected Lagrangian methods; Barrier methods (for inequality constraints); Interior-point methods (for inequality constraints); Sequential linearly constrained programming; Sequential quadratic programming;
4. More topics in optimization: Convexity; Linear programming and the simplex method; Quadratic programming; Duality; Nonlinear least-squares problems; Variational calculus; Nonsmooth optimization; Dynamic optimization and the maximum principle of Pontryagin; Dynamic programming and the Hamilton-Jacobi-Bellman equation; Neural networks and the backpropagation algorithm; Stochastic optimization; Simulated annealing; Genetic algorithms;

Grading procedures: One mid-term examination (30%), one final examination (35%), and homework (35%).