Functional Analysis: Fundamentals
Optimization problems aim at minimizing or maximizing functions over a set. They are used in a wide range of applications, in particular engineering and finance.The objective of the course is to develop modelling skills (writing some real-life applications as optimization problems), to learn important tools of convex analysis, and to study solution methods for some classes of optimization problems:
1) Convexity. Properties of convex and strongly convex functions.
2) Classes of optimization problems.
3) First and second order optimality conditions.
4) Lagrange multipliers and duality.
5) Gradient method.
6) Line searches.
7) Newton and quasi-Newton methods.
8) Subgradient method.
9) Conjugate gradient.
10) Uzawa method
11) Mirror descent and stochastic mirror descent.
12) Cutting plane and bundle methods.
13) Dynamic and dual dynamic programming with cut selection.
14) Implementation of numerical optimization algorithms.
· A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization, SIAM, Philadelphia, 2001.
· D. Bertsimas, J.N Tsitsiklis, Introduction to linear optimization, Athena Scientific, 1997.
· S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2009.
· A. Shapiro, D. Dentcheva, A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2009.
· V. Guigues, Inexact decomposition methods for solving deterministic and stochastic convex dynamic programming equations, arXiv, 2017.
· V. Guigues, Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures, V. Guigues, Mathematical programming, 163(1), 169-212, 2016.
· V. Guigues, Dual Dynamic Programing with cut selection: Convergence proof and numerical experiments, European Journal of Operational Research, 258, 47-57, 2017.