
Akian, M., Gaubert, S., & Hochart, A. (2020). A Game Theory Approach To The Existence And Uniqueness Of Nonlinear PerronFrobenius Eigenvectors. Discret. Contin. Dyn. Syst., 40(1), 207–231.
Abstract: We establish a generalized PerronFrobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear orderpreserving and positively homogeneous map f acting on the open orthant R>0(n). This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a twoperson game that only depends on the behavior of f “at infinity”. In this way, we characterize the situation in which for all alpha, beta > 0, the “slice space” Salpha(beta) :={x is an element of R>0(n) vertical bar alpha x <= f(x) <= beta x} is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations g of f, all the orbits of g are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of f near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors.



Bolte, J., Hochart, A., & Pauwels, E. (2018). Qualification Conditions In Semialgebraic Programming. SIAM J. Optim., 28(2), 1867–1891.
Abstract: For an arbitrary finite family of semialgebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies the MangasarianFromovitz constraint qualification. Using the MilnorThom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential bound is relevant. Our perturbation approach provides a simple protocol to build sequences of “regular” problems approximating an arbitrary semialgebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.



Hochart, A. (2021). Unique Ergodicity of Deterministic ZeroSum Differential Games. Dyn. Games Appl., 11, 109–136.
Abstract: We study the ergodicity of deterministic twoperson zerosum differential games. This property is defined by the uniform convergence to a constant of either the infinitehorizon discounted value as the discount factor tends to zero, or equivalently, the averaged finitehorizon value as the time goes to infinity. We provide necessary and sufficient conditions for the unique ergodicity of a game. This notion extends the classical one for dynamical systems, namely when ergodicity holds with any (suitable) perturbation of the running payoff function. Our main condition is symmetric between the two players and involve dominions, i.e., subsets of states that one player can make approximately invariant.

