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:
- Convexity. Properties of convex and strongly convex functions.
- Classes of optimization problems.
- First and second order optimality conditions.
- Lagrange multipliers and duality.
- Gradient method.
- Line searches.
- Newton and quasi-Newton methods.
- Subgradient method.
- Conjugate gradient.
- Uzawa method
- Mirror descent and stochastic mirror descent.
- Cutting plane and bundle methods.
- Dynamic and dual dynamic programming with cut selection.
- 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.