Cancelled Convex Optimization and Equilibrium Modeling

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.