arXiv subjects

Sergio Mayorga

Publications and source records attributed to Sergio Mayorga.

4 recordsLinked to original sources

arXiv2026

On a discrete max-plus transportation problem

We provide an explicit algorithm to solve the idempotent analogue of the discrete Monge-Kantorovich optimal mass transportation problem with the usual real number field replaced by the tropical (max-plus) semiring, in which addition is defined as the maximum and product is defined as usual addition, with minus infinity and zero playing the roles of additive and multiplicative identities. Such a problem may be naturally called tropical or "max-plus" optimal transportation problem. We show that the solutions to the latter, called the optimal tropical plans, may not correspond to perfect matchings even if the data (max-plus probability measures) have all weights equal to zero, in contrast with the classical discrete optimal transportation analogue, where perfect matching optimal plans in similar situations always exist. Nevertheless, in some randomized situation the existence of perfect matching optimal tropical plans may occur rather frequently. At last, we prove that the uniqueness of solutions of the optimal tropical transportation problem is quite rare.

math.OC↗

arXiv2025

Norm-Constrained Flows and Sign-Based Optimization: Theory and Algorithms

Sign Gradient Descent (SignGD) uses only the coordinate-wise sign of the gradient. We study this method through norm-constrained continuous-time dynamics: at each point, the velocity is chosen to minimize the directional derivative over a unit norm ball. This recovers the sign flow for the $\ell_\infty$ constraint, normalized gradient flow for $\ell_2$, and greedy coordinate directions for $\ell_1$. We formulate the resulting dynamics as a set-valued differential inclusion, prove existence of solutions, and derive an exact energy identity. This identity also gives finite-time convergence under a Polyak-Lojasiewicz inequality expressed in the dual norm. We then connect the canonical set-valued flow with classical Filippov regularization of discontinuous selectors, which gives a precise description of crossing and sliding near switching sets. Motivated by this behavior, we introduce two face-aware SignGD variants, one-hit freeze and two-hit sliding-track. Both methods modify the usual sign direction by damping selected coordinates when a crossing or persistent switching is detected. We derive descent certificates for these damped sign directions and introduce a safeguard that preserves a global linear convergence rate under $\ell_\infty$-smoothness and a Polyak-Lojasiewicz inequality. For the $\ell_1$ flow, we also introduce convex-combination updates on active faces and prove a corresponding linear convergence guarantee. Finally, we analyze mass-aware face restrictions and an inertial SignGD method with restart. Numerical experiments illustrate the certified step rules, the face-aware variants, and the inertial scheme.

math.OC↗

arXiv2021

A note on mean field games of controls with state constraints: existence of mild solutions

We show the existence of "mild solutions" for a first-order mean field game of controls under the state constraint that trajectories be confined in a closed and bounded set in euclidean space. This extends the results of Cannarsa and Capuani to the case of a mean field game of controls. Our controls are velocities and we find that the existence of an equilibrium is complicated by the requirement that they should have enough regularity. We solve this by imposing a small Lipschitz constant on the dependence of the Lagrangian on the joint measure of states and controls, and showing that regular paths can be approximated within the same class of functions despite the constraint.

math.OC↗

arXiv2019

Short time solution to the master equation of a first order mean field game system

The goal of this paper is to show existence of short-time classical solutions to the so called Master Equation of \emph{first order} Mean Field Games, which can be thought of as the limit of the corresponding master equation of a stochastic mean field game as the individual noises approach zero. Despite being the equation of an idealistic model, its study is justified as a way of understanding mean field games in which the individual players'~randomness is negligible; in this sense it can be compared to the study of ideal fluids \cite{gangboberkeleynotes}. We restrict ourselves to potential mean field games but do not impose any monotonicity conditions on the running and initial costs, and we do not require convexity of the Hamiltonian, thus extending the result of \cite{mfgmain} to a considerably broader class of Hamiltonians.

math.AP↗