First-order methods for large-scale market equilibrium computation

Market equilibrium is a solution concept with many applications such as digital ad markets, fair division, and resource sharing. For many classes of utility functions, equilibria are captured by convex programs. We develop simple first-order methods that are suitable for solving these programs for large-scale markets. We focus on three practically-relevant utility classes: linear, quasilinear, and Leontief utilities. Using structural properties of a market equilibrium under each utility class, we show that the corresponding convex programs can be reformulated as optimization of a structured smooth convex function over a polyhedral set, for which projected gradient achieves linear convergence. To do so, we utilize recent linear convergence results under weakened strong-convexity conditions, and further refine the relevant constants, both in general and for our specific setups. We then show that proximal gradient (a generalization of projected gradient) with a practical version of linesearch achieves linear convergence under the Proximal-PL condition. For quasilinear utilities, we show that Mirror Descent applied to a specific convex program achieves sublinear last-iterate convergence and recovers the Proportional Response dynamics, an elegant and efficient algorithm for computing market equilibrium under linear utilities. Numerical experiments show that proportional response is highly efficient for computing an approximate solution, while projected gradient with linesearch can be much faster when higher accuracy is required.