Julian Schwarz
Julian Schwarz

Mathematical Optimization · University of Passau

Julian
Schwarz

I work at the interface of algorithmic game theory and mathematical optimization, mainly on dynamic traffic equilibria with tolls and on equilibrium problems with mixed-integer variables.

I am a research associate at the Chair of Mathematical Optimization (Prof. Dr. Tobias Harks) at the University of Passau. In July 2026 I completed my doctorate in mathematics with a thesis on toll pricing in dynamic equilibrium flows.

julian.schwarz@uni-passau.deORCIDPublicationsCV

The edge beside the text carries a flow over time. Scrolling down moves time forward, scrolling up runs it backwards; the front of the blue band stays at the bottom of your screen, and its width is the flow rate. The network beside the text carries a flow over time. Scrolling down moves time forward, scrolling up runs it backwards. Two commodities, blue and green, travel side by side in a Vickrey network: the width of each band is its flow rate, the distance between the walls of an edge its capacity. Orange containers at the end of an edge hold queues. Starting with the first paper on tolls in the publication list, edges charge tolls (magenta): blue pays them, green takes a detour, and no queue forms.The width of the blue band is the current flow rate, the distance between the walls of each edge its capacity. Orange containers at the end of an edge hold queues. Further down, the flow hands over to a branch-and-cut search for a Nash equilibrium. Slow computer?

News

Research

Generally speaking, I really enjoy areas where combinatorics, optimization, and analysis meet. I am particularly drawn to the foundational side of problems: finding the right concepts for the model at hand and then rigorously proving existence and structural properties of solutions, while paying careful attention to the details.

Dynamic flows and tolls

Dynamic flows describe how traffic, data or goods move through a network over time, with congestion that builds up and dissolves. My doctoral thesis proves a decomposition theorem for dynamic flows, characterizes the flows that tolls can implement as equilibria, and shows that system optimal flows, unlike static ones, are not always implementable. In contrast, for well-behaved single-destination networks, the implementability of system optimal flows is proven.

Generalized Nash equilibrium problems

Generalized Nash equilibrium problems are games in which the strategies a player may choose depend on the choices of the others. I study them with mixed-integer variables, where the players' problems are nonconvex. So far, I have studied them along three questions: how their equilibria can be characterized by convexification, when these equilibria are rational (a prerequisite for exact algorithms), and how branch-and-cut methods compute an equilibrium, exact or approximate, or prove that none exists.

Pricing in nonconvex games and markets

Tolls for dynamic flows are one instance of a more general question: which outcomes can prices on resources enforce as an equilibrium? Players with nonconvex, for example mixed-integer, strategy spaces pay for every unit of a resource they consume, and the question is for which targeted aggregate consumption there are prices that induce an equilibrium with exactly this consumption. One duality-based framework covers this question for general finite-dimensional resource allocation games. If such a game does not admit suitable prices, we study market-clearing prices with minimal regret.

Publications

Journal articles

Conference proceedings

Submitted and preprints

Thesis

Teaching

I taught the exercise classes for these lectures.

Talks

Curriculum vitae

Positions

Education

Service

When I'm not working...

I'm probably biking :D

Julian Schwarz riding a road bike on a mountain road

Contact

Email
julian.schwarz@uni-passau.de
Office
Chair of Mathematical Optimization
University of Passau
ORCID
0000-0003-3451-7473