Vector Space Methods
||Electrical and Computer Engineering
Instance: 2019/2020 - 1S
Cycles of Study/Courses
||No. of Students
||Syllabus since 2015/16
Teaching Staff - Responsibilities
This is an advanced course in functional analysis and infinite dimensional optimization, with applications in least-squares estimation, nonlinear programming in Banach spaces, optimization. The repertoire of analytical tools related to linear spaces provides the students with the facility to investigate new theoretical concepts in electrical engineering specialties
Learning outcomes and competences
Understanding and use of mathematical tools into the resolution of engineering problems.
Pre-requirements (prior knowledge) and co-requirements (common knowledge)
Calculus, Numerical analysis and Linear Algebra
1- An introduction to functional analytic approach to optimization; Finite- versus infinitedimensional
2- Normed linear spaces, open and closed sets; convergence; continuity; Banach
spaces, Complete subsets, Quotient spaces, Denseness and Separability
3- Fixed points of transformations on Banach Spaces -- Applications to solutions of
ordinary differential and integral equations
4- Hilbert Spaces -- The Projection Theorem; Orthogonal Complements; Gram-Schmidt
Procedure; Minimum distance to a convex set
5- Hilbert Spaces of random variables and stochastic processes; Least-squares
6- Dual Spaces. The Hahn-Banach Theorem, with applications to minimum norm
7- Linear operators and adjoints
8- Optimization of functionals -- General results on existence and uniqueness of an optimum
9- Optimization of functionals. Gateaux and Frechet derivatives. Extrema; Euler-
Lagrange equations; Min-Max Theorem in Game Theory.
10- Constrained optimization of functionals: Global theory; Convex-concave functionals,
conjugate functionals, dual optimization problems, Lagrange multipliers, sufficiency;
sensitivity, duality; applications .
11- Constrained Optimization; Equality and Inequality Constraints; Kuhn-Tucker´Theorem in infinite dimensions
12- Other related topics (as time permits)
Luenberger, David G.; Optimization by Vector Space methods
Polak, E; Optimization: Algorithms and Consistent Approximations
, Springer, New York, 1997. ISBN: 0-387-94971-2
Boyd, S. and Vandenberghe, L; Convex Optimization
, Cambridge University Press, 2005. ISBN: 0 521 83378 7
Varaiya, Pravin; Lecture notes on optimization
, e-book, http://paleale.eecs.berkeley.edu/~varaiya/papers_ps.dir/NOO.pdf
Teaching methods and learning activities
There will be expository lectures in the end of which a list of problems are proposed. Such lectures are followed by discussion classes to treat problems assigned on the subject.
Distributed evaluation without final exam
Amount of time allocated to each course unit
Eligibility for exams
90% of the homework with grade greater or equal to 10.
Calculation formula of final grade
0,80 * homework + 0,20 * class participation
Examinations or Special Assignments
Students will have to do different homeworks that should be returned within a
week after being assigned.
With an extra project on Optimization.
Classes can be in Portuguese if no foreigners are enrolled.