Contractivity and linear convergence in bilinear saddle-point problems: An operator-theoretic approach

We study the convex-concave bilinear saddle-point problem $\min_x \max_y f(x) + y^\top Ax - g(y)$, where both, only one, or none of the functions $f$ and $g$ are strongly convex, and suitable rank conditions on the matrix $A$ hold. The solution of this problem is at the core of many machine learning tasks. By employing tools from operator theory, we systematically prove the contractivity (in turn, the linear convergence) of several first-order primal-dual algorithms, including the Chambolle-Pock method. Our approach results in concise and elegant proofs, and it yields new convergence guarantees and tighter bounds compared to known results.

Designing Optimal Personalized Incentive for Traffic Routing using BIG Hype algorithm

We study the problem of optimally routing plug-in electric and conventional fuel vehicles on a city level. In our model, commuters selfishly aim to minimize a local cost that combines travel time, from a fixed origin to a desired destination, and the monetary cost of using city facilities, parking or service stations. The traffic authority can influence the commuters' preferred routing choice by means of personalized discounts on parking tickets and on the energy price at service stations. We formalize the problem of designing these monetary incentives optimally as a large-scale bilevel game, where constraints arise at both levels due to the finite capacities of city facilities and incentives budget. Then, we develop an efficient decentralized solution scheme with convergence guarantees based on BIG Hype, a recently-proposed hypergradient-based algorithm for hierarchical games. Finally, we validate our model via numerical simulations over the Anaheim's network, and show that the proposed approach produces sensible results in terms of traffic decongestion and it is able to solve in minutes problems with more than 48000 variables and 110000 constraints.