Numerical Optimization

Recommended Prerequisites

Basic courses in (Differential and Integral) Calculus, Linear Algebra and Numerical Analysis.

Teaching Methods

The classes are essentially of expository style and include examples and exercises to apply the material being taught.

There are two types of grading: during the semester or by final exam. During the semester there are two mid-term exams (50-75% of the final grade) and a set of homework assignments (50-25% of the final grade) given every two or three other weeks and handed in individually. The exercises in the homework assignments are mathematical problems or short numerical tasks. The final exam option consists of a single exam (100% of the final grade).

Extensive tutorial time is offered to the students to support the solution of the homework assignments and preparation for the various exams.

Learning Outcomes

The main goal is teaching the main numerical methods for unconstrained and constrained non-linear optimization, their motivation, their numerical features and their convergence properties. One also aims at studying the theory of constrained optimization and the corresponding duality theory.

The course aims at developing the following skills: knowledge of mathematical results; ability to generalize and abstract; logic thinking; competence in using computational tools. On the personal level it also allows to develop self-learning skills and independent thinking.

Work Placement(s)

No

Syllabus

(1) Numerical methods for unconstrained non-linear optimization (direct-search methods, line-search methods, steepest descent method, modified Newton's method, trust-region methods, global and global-local proprieties of the several methods).

(2) Theory of constrained non-linear optimization (constraint qualifications; necessary and sufficient conditions; duality theory). The particular cases of linear and quadratic programming.

(3) Numerical methods for constrained non-linear optimization (quadratic penalty method; augmented Lagrangian method; sequential quadratic programming method and merit functions; interior point method).

Assessment Methods

Continuous assessment
There are two types of grading: during the semester or by final exam. During the semester there are two mid-term exams (50-75% of the final grade) and a set of homework assignments (50-25% of the final grade) given every two or three other weeks and handed in individually. The exercises in the homework assignments are mathematical problems or short numerical tasks.: 100.0%

Final assessment
Exam: 100.0%

Bibliography

J. Nocedal, S.J. Wright, Numerical Optimization, segunda edição, Springer, 2006.

A.R. Conn, K. Scheinberg, L.N. Vicente, Introduction to Derivative-Free Optimization, MPS-SIAM Book Series on Optimization, SIAM, 2009.

J.E. Dennis, R.B. Schnabel, Numerical methods for Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.

I. Griva, S.G. Nash, A. Sofer, Linear and Nonlinear Optimization, segunda edição, SIAM, 2009.