Math 271A: Numerical Optimization

Fall 2021

Problems in all areas of mathematics, applied science, engineering, economics, medicine and statistics can be posed as mathematical optimization problems. An optimization problem begins with a set of independent variables, and often includes conditions or restrictions that define acceptable values of the variables. Such restrictions are known as the constraints of the problem. The other essential component of an optimization problem is a single measure of ``goodness'', termed the objective function, which depends in some way on the variables. The solution of an optimization problem is a set of allowed values of the variables for which the objective function assumes its ``optimal'' value. In mathematical terms, this usually involves maximizing or minimizing. Numerical optimization problems arise in many applications, ranging from the optimal control of the Mars Lander, to the optimal design of the hull of the New Zealand 2000 (winning) entry to the Americas Cup yacht race.

This class is intended as an introduction to the design and analysis of algorithms for numerical optimization. It is suitable for graduate students in the applied and natural sciences who want to develop an understanding of practical methods for optimization. If time permits, discussion will include some case studies involving real problems.

Specific topics will include unconstrained optimization; optimality conditions; Newton and quasi-Newton methods; the solution of nonlinear equations; constrained optimization; the Karush-Kuhn-Tucker conditions; linear, quadratic programming and nonlinear programming; convex programming; penalty- and barrier-function methods; interior-point methods; KKT systems and their numerical solution; augmented Lagrangian methods, sequential quadratic programming (SQP) methods.

Lecture notes are available to registered students from the link below. Some homework assignments will require the use of the interactive matrix package Matlab, although no extensive prior knowledge of Matlab is assumed. Matlab enables the student to concentrate on the fundamental ideas of numerical optimization without becoming distracted by the rigors of mental arithmetic. (``It is unworthy of excellent men to lose hours like slaves in the labour of calculation.''---Leibniz.)

Class Location:

Grading options:

• Letter grade: The grade will be based on a final examination: Wednesday, December 8, 7:00p-9:59p
  The final will be a "closed-book" examination based on homework problems assigned during the quarter.
• SU grade: Students need not take the final examination, but must make a "fair attempt" at the homework assignments.

Class notes and homeworks: