Numerical Analysis and Optimization
Period of duration of course
The course is suitable for those students who have had a first introductory course on numerical methods but have not yet been exposed to systematic courses on Numerical Linear Algebra and Optimization.
Part I, Computational Methods of Linear Algebra
Review of basic facts from linear algebra.
Matrix norms. Singular value decomposition (SVD).
Stability and conditioning in Numerical Linear Algebra. Numerical rank.
Matrix Factorizations (LU, Cholesky, QR...).
Direct and Iterative methods for solving linear systems.
Least squares problems. Moore-Penrose pseudoinverse.
Computation of eigenvalues and eigenvector of matrices.
Part II, Methods of Numerical Optimization
-Gradient descent, Newton and Quasi-Newton methods, Nesterov's method.
-Method of Lagrange multipliers, augmented Lagrangian methods.
-Interior point methods.
-The ADMM method.
The goal of this course is to provide the students the basic tools of numerical linear algebra and of optimization (both constrained and unconstrained).
The emphasis will be on the fundamental concepts (in particular those of stability and conditioning of problems) and on the algorithmic aspects.
J. Demmel, Applied Numerical Linear ALgebra, SIAM, 1997.
J. Nocedal and S. Wright, Numerical Optimization, Springer, 1998.
Additional references will be provided in the course of the lectures.