MA662 Stochastic Optimization


Course Catalog Description

Objective

The methodology of optimal operation of stochastic systems is known as stochastic programming or stochastic optimization. It is a rapidly developing part of applied mathematics based on probability theory, statistics, mathematical optimization and optimal control. The main purpose of this course is to introduce students to the basic models of optimal decisions under uncertainty and risk. We shall discuss various modeling approaches, some basic properties of the models, as well as numerical methods for their solution. The theory will be illustrated by applied problems.

Instructors

Professor Email Office
Darinka Dentcheva darinka.dentcheva@stevens.edu

More Information

Course Outcomes

1. Ability to formulate an appropriate optimization model for a decision making problem involving uncertain parameters.
2. Knowledge of the mathematical models of stochastic optimization; their structure, advantages, and limitations.
3. Knowledge of the mathematical models of risk-averse optimization and the relations between them.
4. Knowledge of the numerical approaches for solving problems with constraints on probability of events, stochastic-order constraints, two- and multi-stage problems.
5. Understanding of time-consistency for dynamic risk.
6. Understanding the statistical meaning of the optimal value and solution for stochastic optimization problems based on sampled data.

Course Materials

There is no required textbook; lecture notes will be distributed in class. The following are supplementary books:

  1. A. Shapiro, D. Dentcheva, A. Ruszczynski: Lecture Notes on Stochastic Programming Modeling and Theory, SIAM and MPS, Second Edition 2014.The first edition is available online.
  2. A. Ruszczynski and A. Shapiro, Stochastic Programming, Handbook in Operations Research and Management Science, Elsevier Science, Amsterdam, 2003
  3. Andras Prekopa, Stochastic Programming, Kluwer Academic Publishers, Netherlands, 1995


Grading

Grading Policies

A student's course grade will be based on 7-8 assignments.


Lecture Outline

Date Topic Reading
Week 1 Modeling issues in presence of uncertainty and risk.
Week 2 Optimization problem with probabilistic (chance) constraints: properties.
Week 3 Numerical solution of optimization problems with probabilistic constraints.
Week 4 Calculating bounds on probability of events.
Week 5 Stochastic optimization with recourse. Properties of the expected value functional. The structure of stochastic models with recourse.
Week 6 Numerical methods for solving two-stage problems.
Week 7 Multistage stochastic problems and their properties.
Week 8 Numerical techniques for multistage models.
Week 9 Risk-averse optimization:stochastic-order constraints.
Week 10 Numerical methods for problem with stochastic-order constraints
Week 11 Optimization problems with coherent measures of risk.
Week 12 Dynamic measures of risk. Time consistency.
Week 13 Optimization of dynamic measures of risk.
Week 14 Statistical aspects of stochastic optimization models.