Numerical Methods in Optimization and Differential Equations

The study of numerical methods for solving optimization problems and differential equations in order to provide the students with the fundamental knowledges for their analysis, the implementation in a computer and the application to specific problems.

1. Numerical solution of differential equations. (15 h.)
1.1. Numerical solution of initial value problems for O.D.E. (14 h.)
1.1.a. Basic methods: explicit and implicit Euler, theta-method and
mid-point rule. (2 h.)
1.1.b. Concepts of consistence, stability, convergence, order and
numerical stability. Stiff problems. (5 h.)
1.1.c. Runge-Kutta methods, linear multistep methods: description
and properties. (7h)
1.2. Numerical solution of the boundary value problem for the
ordinary second order linear equation: a finite differences scheme:
description and analysis. (1 h.)

2. Numerical methods in optimization. (13h)
2.1. Numerical methods in unconstrained optimization. (9 h.)
2.1.a. Existence and uniqueness of solution: convex sets and
convex functions, optimality conditions. (2 h.)
2.1.b. One-dimensional optimization: Armijo, Goldstein and Wolfe-Powell rules. (2 h.)
2.1.c. Gradient and conjugate gradient methods. (5 h.)
2.2. Numerical methods in constrained optimization. (3 h.)
2.2.a. Existence and uniqueness of solution: Lagrange multipliers
and optimality conditions. (2 h.)
2.2.b. Penalty methods. (1 h.)
2.3 Discrete linear least squares approximation.
Existence and uniqueness of solution: normal equations. (1 h.)

Basic references on numerical methods for differential equations:
E. Hairer, S. P. Nørsett, G. Wanner (1987): Solving Ordinary Differential Equations I. Non-stiff Problems. Springer. (Available online).

Basic references on numerical methods in optimization:
J. Viaño, M. Burguera (2013): Lecciones de Métodos Numéricos. 4.- Optimización. Andavira Editora. Santiago de Compostela.
W. Sun, Y. Yuan (2006): Optimization Theory and Methods. Springer. (Available online).

Complementary references on numerical methods for differential equations:
E. Hairer, G. Wanner (1991): Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer. (Available online).
J. D. Lambert (1991): Numerical Methods for Ordinary Differential Systems. Wiley.
J. C. Butcher (2003): Numerical Methods for Ordinary Differential Equations. Wiley.
M. Crouzeix, A. L. Mignot (1989): Analyse Numérique des Équations Differentielles. Masson.

Complementary references on numerical methods in optimization:
J. E. Dennis, R. B. Schnabel (1983): Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall.
D. G. Luenberger (1973): Introduction to Linear and Nonlinear Programming. Addison-Wesley.
D. P. Bertsekas (1995): Nonlinear programming. Athena Scientific.
J. Nocedal, S. J. Wright (1999): Numerical Optimization. Springer-Verlag. (Available online).

Reference on numerical methods:
W. Gander, M. J. Gander, F. Kwok (2014): Scientific computing – An introduction using MAPLE and MATLAB. Springer. (Available online).
S. R. Otto, J. P. Denier (2005): An Introduction to Programming and Numerical Methods in Matlab. Springer. (Available online).
R. L. Burden, J. D. Faires (1998): Análisis Numérico. ITP Thomson.
E. Isaacson, H. B. Keller (1994): Analysis of Numerical Methods. Dover.
D. Kincaid, W. Cheney (1994): Análisis numérico: las matemáticas del cálculo científico. Addison-Wesley Iberoamericana.

The skills detailed in the Memoria de Verificación de Título do Grao en Matemáticas
(http://www.usc.es/export9/sites/webinstitucional/gl/servizos/sxopra/mem…) will be exercised.

The teaching methodology will be based on lectures where the theoretical concepts of the subject will be presented. These contents will be put into practice in computer labs where the methods previously presented will be programmed and where a selection of exercises from the worksheets will be solved.

The subject will have a web page on the virtual campus where various documents and activities will be posted. This platform will also be used to communicate with the students.

If it is necessary to hold a virtual session by videoconference, the Teams platform will be used.

The accomplishment of the objectives, both in terms of contents and skills, will be graded through a final exam and continuous evaluation.

In the final exam (EF, maximum of 10 points), to be held on the official date assigned by the faculty, the theoretical concepts acquired, the ability to solve questions and problems (ET, maximum of 7.5 points) as well as the programming skills (EP, maximum of 2.5 points) will be evaluated. To be more precise
EF = ET + EP.

The grade related to continuous evaluation (AC, maximum of 10 points) will be calculated based on one intermediate tests/activity. This activity will include exercises similar to those in the assigned worksheets and programming a numerical method.

The final grade (CF) will be obtained after calculating the maximum between EF and the weighted average between EF (75%) and AC (25%). To be more precise:
CF = max {EF, 0.75 * EF + 0.25 * AC}

The final grade in the second opportunity will be calculated with the following formula
CF = max {EF2, 0.75 * EF2 + 0.25 * AC}
where EF2 will be the grade obtained in the second-chance exam (which will have the same characteristics as the first one).

The assessment system for the different groups will be the same.

The same assessment criteria will be applied to students who repeat the course.

The grade of "no presentado" will be awarded to those students who do not carry out any evaluable activity.

For the exclusive effect of granting the Honor Registration qualification, not only the final numerical grade will be taken into account, but also the continuous evaluation.

For cases of fraudulent performance of exercises or tests, the provisions of the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” may be applied.

Total hours of work with the teacher: 58h.
- Lectures: 28h.
- Interactive laboratory classes: 28h.
- Classroom tutoring: 2h.

Total hours of personal work: 92h.
- Individual or group autonomous study: 42h
- Programming / experimentation or other computer / laboratory work: 35h
- Writing of exercises, conclusions or other works: 10h
- Recommended readings and activities with bibliographic support: 5h

The total number of estimated hours to pass the subject is 92h + 58h = 150h.

- Diary study of contents covered in the class, supplemented by notes given by the teacher.
- Use of the tutorials to solve all sorts of doubts about the matter.
- Resolution of the worksheets and search of others in the recommended literature.
- Programming of the proposed algorithms, within the marked delays.