ENSTA ParisTech - Cours de M2

	University Paris-Saclay
	Master ``Optimization''
	Academic Year 2018/2019
	First term's advanced courses
	Stochastic Optimization

Professors

Pierre Carpentier and Vincent Leclère

Goals

The course presents both theoretical and numerical aspects of decision problems with uncertainty, where one sets a probabilistic framework in order to minimize the expectation of a cost. Two directions are explored:

we investigate the so-called "open-loop" situation, that is, the case where decisions do not depend on information available for the problem, and we thoroughly study the stochastic gradient method and its variants,
we also study "closed-loop" optimization problems, that is, the case where decisions are based on partial information (often corresponding to measurements made in the past when facing an unknown future).

Such problems are of course well-motivated by decision problems in the industry. They also have a deep mathematical content, especially in the dynamic case when only the past information is available. In this setting the decision is a function in a high dimensional space and therefore the numerical aspects also are challenging.
This course is part of the M2 Optimization program (Paris-Saclay University).

Structure

The course takes place at ENSTA on Wednesday, from 09:00 to 12:00 and from 14:00 to 16:00, and is given in English. Getting to ENSTA

Lesson 1 (November 21, room 2.2.36)
- 09:00 - 12:00 (P. Carpentier)
  Issues in decision making under uncertainty. Slides
- 14:00 - 16:00 (V. Leclère)
  Convex analysis and probability tools for stochastic optimization - Part I.

Lesson 2 (November 28, room 2.2.36)
- 09:00 - 12:00 (P. Carpentier)
  Stochastic gradient method overview. Slides
- 14:00 - 16:00 (V. Leclère)
  Convex analysis and probability tools for stochastic optimization - Part II.

Lesson 3 (December 05, room 2.2.36)
- 09:00 - 12:00 (P. Carpentier)
  Generalized stochastic gradient method. Slides
- 14:00 - 16:00 (V. Leclère)
  Stochastic Programming. The two-stage case.

Lesson 4 (December 12, room 2.2.36)
- 09:00 - 12:00 (V. Leclère)
  Scenario decomposition: L-Shaped and Progressive Hedging methods.
- 14:00 - 16:00 (P. Carpentier)
  Stochastic gradient method with constraint in expectation and applications. Slides

Lesson 5 (December 19, room 2.2.36)
- 09:00 - 12:00 (P. Carpentier)
  Discretization issues of general stochastic optimization problems. Slides
- 14:00 - 16:00 (V. Leclère)
  Bellman operators and Stochastic Dynamic Programming.

Lesson 6 (January 09, room 2.2.36)
- 09:00 - 12:00 (V. Leclère)
  The Stochastic Dual Dynamic Programming (SDDP) approach.
- 14:00 - 16:00 (P. Carpentier)
  Decomposition approaches for large scale stochastic optimization problems. Slides

Evaluation (January 16, room 2.2.36).
- 09:00 - 12:00 (P. Carpentier)
  Articles presentation - Session 1.
- 14:00 - 16:00 (V. Leclère)
  Written exam.
Evaluation (January 21, room 2.2.36).
- 09:00 - 12:00 (P. Carpentier)
  Articles presentation - Session 2.