Numerical Methods and Optimization

year 2015/2016

The course introduces some of the main techniques and methods for the solution of numerical problems. These methods often require the joint exploitation of the typical techniques of numerical analysis and of optimization algorithms. We show some of the main situations in which optimization methods are applied to solve problems of numerical analysis and some of the main situations in which the techniques of numerical analysis are essential to solve optimization problems. We also show the application of these methods to some specific problems chosen, for instance, in the following areas: regression and parameter estimation in statistics, approximation and data fitting, machine learning, data mining, image and signal reconstruction, economic equilibria and finance.

The program below is a preliminary version, which will be periodically updated according to the actual lectures.


(A more detailed program will be soon available through the record of lectures)

  1. Topology and calculus background

  2. Linear algebra background

  3. Convex functions, convex sets and optimization problems

  4. Optimality conditions for unconstrained optimization

  5. Direct and iterative methods for linear systems

  6. Iterative methods for nonlinear systems

  7. Solution methods for unconstrained optimization

  8. The least-squares problem

  9. Iterative methods for computing eigenvalues

  10. Optimality conditions for constrained optimization

  11. Solution methods for constrained optimization

  12. The fast Fourier transform