# NMAK23003U Convex Optimization and Equilibrium Modeling

Volume 2023/2024
Education

MSc Programme in Actuarial Mathematics
MSc Programme in Mathematics-Economics

Content

Equilibrium modeling has proved to be very effective in solving various problems in economics and transportation. Examples include finding traffic equilibria, spatial price equilibria, and Cournot-Nash equilibria. One important area of application involves energy markets, where equilibrium models may be used to monitor the abuse of market power and predict the effects of decarbonization policies.

This course starts with an introduction to convex optimization. Convex optimization problems arise in a broad number of fields and applications. After introducing the concepts of convex sets and convex functions, we are able to recognize certain optimization problems as being convex. We proceed to cover duality theory and optimality conditions for these problems. Finally, we discuss solution methods for unconstrained and constrained convex optimization problems.

The second part of the course focuses on equilibrium modeling. A short introduction to game theory allows to understand the notion of a Nash equilibrium. We furthermore consider complementarity problems and variational inequalities, with an emphasis on the relationship between these problems and convex optimization.

The content of the course is as follows:

A. Convex optimization:

• A1. Convex sets.
• A2. Convex functions.
• A3. Convex optimization problems.
• A4. Duality and optimality.

B. Equilibrium modeling:

• B1. Nash equilibria.
• B2. Complementarity problems.
• B3. Variational inequalities.
Learning Outcome

Knowledge:

• Properties of convex sets, convex functions, and convex optimization problems
• Solution methods for convex optimization problems
• Definitions of a Nash equilibrium, complementarity problem, and variational inequality

Skills:

• Classify convex optimization problems
• Recognize and formulate convex optimization and equilibrium problems
• Implement and solve a given optimization or equilibrium problem using appropriate software
• Understand and reproduce the proofs presented in the course

Competences:

• Explain how to exploit duality theory and optimality conditions for convex optimization problems in the design of a solution method
• Describe the differences and relationships between convex optimization problems, complementarity problems, and variational inequalities
• Formulate, implement and solve a practical problem and justify the formulation and solution method
Operations Research 1 (OR1) or similar is required.

Academic qualifications equivalent to a BSc degree is recommended.
2 x 2 hours of lectures and 1 x 2 hours exercises/project work per week for 7 weeks.
• Category
• Hours
• Lectures
• 28
• Preparation
• 133
• Theory exercises
• 14
• Project work
• 30
• Exam
• 1
• Total
• 206
Written
Individual

Individual written feedback will be given on the mandatory assignment.

Credit
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
Type of assessment details
Without preparation time
Exam registration requirements

Approval of the project report is a prerequisite for enrolling for examination.

Aid
Written aids allowed
Marking scale