2nd Copenhagen School of Stochastic Programming

Copenhagen skyline

Welcome to the 2nd Copenhagen School of Stochastic Programming. This is a PhD course that provides a practical and research-oriented introduction to selected topics on stochastic programming, a mathematical framework for decision-making in the presence of uncertainty. The goal is that of providing students the necessary concepts and hands-on experience to begin using these methodology in research and practice. This course is part of a broader initiative named "Nordic Courses on Stochastic Programming" (NCSP) coordinated with NTNU (Trondheim, Norway) and NHH (Bergen, Norway) which aims to provide periodic courses on the topic. The most recent courses were held at the University of Copenhagen in the Spring 2022, at NHH in the Fall 2022 and at NTNU in the Fall 2023.

The course will take place at the University of Copenhagen from Tuesday June 25th 2024 to Friday June 28th (inclusive). This is the week immediately before the EURO 2024 conference (which will also take place in Copenhagen, at the Technical University of Denmark).

The course comprises lectures, practical exercises, and a limited number of student presentations. The lectures will be given by a number of international experts on the topic, namely Alois Pichler, Asgeir Tomasgard, Francesca Maggioni and Michal Kaut (see below) in addition to the local organizers Trine Boomsma and Giovanni Pantuso. Getting started with working on Stochastic Programming topics can sometimes be challenging as it requires an understanding of elements of (at least) optimization, probability and statistics. Therefore, the course offers introductory-level lectures on selected topics. These have the scope of providing the students the ability to immediately start working with such topics. We will start by formalizing decision problems under uncertainty as mathematical optimization problems and analyzing their fundamental mathematical properties. Following we will discuss how to deal with problems characterized by continuous distributions of the random data or by very large discrete distributions. Particularly, we will discuss how to approximate problems using so called scenario generation techniques and how to derive bounds on the true (non-approximated) problem. Furthermore, we will introduce problems characterized by multiple time horizons (e.g., strategic and operating) with their respective random phenomena. We will continue by introducing problems where decisions have an impact on the uncertainty and methods to solve these. Finally, we will discuss how to model problems where the decision maker can be characterized as risk-averse.

Upon registration, students can choose (but it is not mandatory) whether they would like to present a topic based on their research. In this case they can upload an abstract (max 2 pages). Approximately 10 abstracts will be selected for presentation. Student presentations have the scope of complementing the lectures. Therefore, selection is based on coherence with the topics presented by the lectures as well as on quality.

The course is primarily aimed at PhD students who require a solid introduction to decision-making under uncertainty. Nevertheless, it is open to anyone who would like to approach the subject, ranging from master's students to more experienced researchers and professionals. A participant can earn a certificate of 5 ECTS (see below for details). The course is open to students from all over the world. The registration fee includes participation at the course, material and coffee breaks.



Connected to the PhD School there is a Special Issue of the journal Computational Management Science entitled "Stochastic Optimization: Methodological Advancements and Modern Applications". Selected abstracts (not necessarily presented at the course) will be invited to submit a full paper for the special issue. Nevertheless, submission is open to a more general audience. Articles will be reviewed according to the journal standard. Submissions are possible until December 31st 2024.





The lectures will be given by four guest lecturers, Francesca Maggioni, Alois Pichler, Asgeir Tomasgard and Michal Kaut in addition to two local lecturers, Trine Boomsma and Giovanni Pantuso.

  • Francesca Maggioni is Associate Professor of Operations Research at the University of Bergamo (Italy). Her research interests concern both methodological and applicative aspects for optimization under uncertainty. From a methodological point of view, she has developed different types of bounds and approximation for stochastic, robust and distributionally robust multistage optimization problems. She applies these methods to solve problems in logistics, transportation, energy production and pension funds. Francesca is currently the chair of the EURO Working Group on Stochastic Optimization and of the the AIRO Thematic Section of Stochastic Programming. She is Associate Editor of the journals: Computational Management Science (CMS), EURO Journal on Computational Optimization (EJCO), TOP and International Transactions in Operational Research (ITOR).

  • Alois Pichler is a Professor at the Faculty of Mathematics at TU Chemnitz (Germany). Alois' research interest include optimization under uncertainty, risk theory, financial mathematics, statistics and probability theory. He is co-author of the book Multistage Stochastic Optimization (Springer) and his research is published in some of the leading OR journals, including Operations Research, Mathematics of Operations Research, SIAM Journal on Optimization, Computational Management Science. In addition he serves as a committee member for the Stochastic Programming Society, as associate editor for Mathematical Programming and SIAM rev, as a member of the EURO Working Group on Stochastic Optimization and editor for Mathematical Finance.

  • Asgeir Tomasgard is a Professor at the Norwegian University of Science and Technology (Norway). Asgeir's research focuses in particular on decision problems under uncertainty emerging in energy systems. In particular, he studies the interplay of modeling details in problems with long planning horizons (e.g., investments and capacity expansion) and fine short-run operating scenarios. Asgeir currently serves as centre director in Norwegian Centre for Energy Transition Strategies and director of NTNU Energy Transition. In addition, Asgeir serves as a board member in Norsk Klimastiftelse and has been in the management board of the European Working Group for Stochastic Optimization since 2013.

  • Michal Kaut is a researcher at SINTEF, one of Europe’s largest independent research organizations based in Norway. Michal uses optimization techniques including stochastic programming to address real-life problems. Michal's research has focused on approximation methods (i.e., scenario-generation methods) for stochastic programs. He has developed several methods for discretizing continuous distributions depending on the available information. His research is published in some of the leading optimization journals, including EJOR, Computational Management Science, Computational Optimization and Applications, and INFORMS Journal on Computing. Furthermore, computer code is available for some of the methods he has developed.










The course will offer lectures on the following topics:

  • Introduction to Stochastic Programming (Trine Boomsma)
  • Stochastic Programs with Multiple Time Horizons (Asgeir Tomasgard)
  • Bounding techniques in optimization under uncertainty (Francesca Maggioni)
  • Scenario Generation Methods (Michal Kaut)
  • Risk-Aversion in Stochastic Programming (Alois Pichler)
  • Stochastic Programs with Decision-Dependent Uncertainty (Giovanni Pantuso)

The preliminary schedule of the PhD School's activities is as follows. Each day consists of approximately three hours of lectures in the morning and approximately three hours of lectures or student-presentations in the afternoon.

    Tuesday June 25th
    08:30-09:30 Registrations
    09:30-09:45 Welcome session and introduction to the PhD course
    10:00-12:00 Trine K. Boomsma: An Introduction to Stochastic Programming
    12:00-14:00 Lunch break
    14:00-17:00 Student presentations
    Wednesday June 26th
    09:00-12:00 Speaker: Topic
    12:00-14:00 Lunch break
    14:00-17:00 Student presentations
    Thursday June 27th
    09:00-12:00 Speaker: Topic
    12:00-14:00 Lunch break
    14:00-17:00 Speaker: Topic
    Friday June 28th
    09:00-12:00 Speaker: Topic
    12:00-14:00 Lunch break
    14:00-17:00 Speaker: Topic
    17:00-17.30 Closing session



    Intended Learning Outcome and Prerequisites

    The students will become well acquainted with the theory of stochastic programming and the challenges involved when applying stochastic programming in practice. Particularly, upon completion of the course, the students will be able to

    • formulate two-stage and multi-stage stochastic programs
    • account for both risk-neutral and risk-averse decision makers
    • account for time horizons with different resolutions
    • account for the possible influence of decisions on the random outcomes
    • approximate stochastic programs by generating scenarios from a possibly continuous probability distribution in order to obtain practical solutions
    • derive bounds on their optimal objective values

    Prerequisites for a successful participation in the course are a solid understanding of linear programming theory and some knowledge of probability theory (e.g., understanding what probability distributions are for both continuous and discrete random variables, understanding what expectations are). Knowledge of a linear/mixed-integer programming solver is also beneficial.



    There will be a course dinner on June 27th. The dinner will take place at a restaurant in town. Participation at the dinner is optional and requires an additonal fee of 500 DKK. It can be purchased upon registration.














    The course will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the course venue.

    Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

    A journey planner in English is available. More information on the "find us" webpage.

    Students will need to arrange their own accommodation. Here is a list of good options for budget accommodation:



















    Registration closed on May 15th.

    In case registered participants cannot attend, the registration can be cancelled by sending an email to the organizers. The registration fee can be refunded until May 31st.



    The course grants 5 ECTS upon succesful complition of the course. Particularly, in order to obtain an ECTS certificate, the participant needs to:

    • Attend all the lectures and student presentations
    • Deliver an essay on one of the topics of the course

    Further instructions will follow during the course.



    Upon request, the organizers may provide letters of invitation for the purpose of obtaining a visa. In order to obtain a letter of invitation the following conditions must hold:

    • The student must have registered and paid for the course
    • The request must arrive from an institutional email (requests from personal email addresses such as gmail will be discarded)
    • The student must provide proof of enrollment in a PhD program (in English) and the contact of the academic supervisor.

    Please contact the organizers to obtain such letter.