Numerical Analysis and Optimization
Prerequisiti
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.
Programma
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
Unconstrained optimization:
-Gradient descent, Newton and Quasi-Newton methods, Nesterov's method.
-Globalization techniques.
Constrained optimization:
-Method of Lagrange multipliers, augmented Lagrangian methods.
-Interior point methods.
-KKT systems.
-The ADMM method.
Obiettivi formativi
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.
Riferimenti bibliografici
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.