Numerical Analysis and Optimization
Prerequisiti
An excellent knowledge of linear algebra is required. It would be desirable to have some familiarity with numerical computing and scientific programming.
The course is recommended for 1st year PhD students but is accessible to Master's students.
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.
-LASSO, Basis Pursuit, etc.
Obiettivi formativi
To provide students with the theoretical and algorithmic tools to solve linear algebra and optimization problems.
Riferimenti bibliografici
J. Demmel, Applied Numerical Linear Algebra, SIAM, 1997.
A. Bjorck, Numerical Methods in Matrix Computation, Springer, 2015.
J. Nocedal and S. Wright, Numerical Optimization, Springer, 1999.