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Author Bolte, J.; Hochart, A.; Pauwels, E.
Title Qualification Conditions In Semialgebraic Programming Type
Year (up) 2018 Publication Siam Journal On Optimization Abbreviated Journal SIAM J. Optim.
Volume 28 Issue 2 Pages 1867-1891
Keywords constraint qualification; Mangasarian-Fromovitz; Arrow-Hurwicz-Uzawa; Lagrange multipliers; optimality conditions; tame programming
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 Mangasarian-Fromovitz constraint qualification. Using the Milnor-Thom 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.
Address [Bolte, Jerome] Univ Toulouse 1 Capitole, Toulouse Sch Econ, Toulouse, France, Email: jerome.bolte@tse-fr.eu;
Corporate Author Thesis
Publisher Siam Publications Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1052-6234 ISBN Medium
Area Expedition Conference
Notes WOS:000436991600036 Approved
Call Number UAI @ eduardo.moreno @ Serial 882
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Author Akian, M.; Gaubert, S.; Hochart, A.
Title A Game Theory Approach To The Existence And Uniqueness Of Nonlinear Perron-Frobenius Eigenvectors Type
Year (up) 2020 Publication Discrete And Continuous Dynamical Systems Abbreviated Journal Discret. Contin. Dyn. Syst.
Volume 40 Issue 1 Pages 207-231
Keywords Nonlinear eigenproblem; nonexpansive map; Hilbert's projective metric; hypergraph; zero-sum stochastic game
Abstract We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving 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 two-person 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” S-alpha(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.
Address [Akian, Marianne; Gaubert, Stephane] Ecole Polytech, INRIA, CNRS, Inst Polytech Paris, F-91128 Palaiseau, France, Email: marianne.akian@inria.fr;
Corporate Author Thesis
Publisher Amer Inst Mathematical Sciences-Aims Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1078-0947 ISBN Medium
Area Expedition Conference
Notes WOS:000496748500009 Approved
Call Number UAI @ eduardo.moreno @ Serial 1075
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Author Hochart, A.
Title Unique Ergodicity of Deterministic Zero-Sum Differential Games Type
Year (up) 2021 Publication Dynamic Games And Applications Abbreviated Journal Dyn. Games Appl.
Volume 11 Issue Pages 109-136
Keywords Differential games; Hamilton-Jacobi equations; Viscosity solutions; Ergodicity; Limit value
Abstract We study the ergodicity of deterministic two-person zero-sum differential games. This property is defined by the uniform convergence to a constant of either the infinite-horizon discounted value as the discount factor tends to zero, or equivalently, the averaged finite-horizon 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.
Address [Hochart, Antoine] Univ Adolfo Ibanez, Fac Ingn & Ciencia, Diagonal Las Torres 2640, Santiago, Chile, Email: antoine.hochart@gmail.com
Corporate Author Thesis
Publisher Springer Birkhauser Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 2153-0785 ISBN Medium
Area Expedition Conference
Notes WOS:000527444200001 Approved
Call Number UAI @ eduardo.moreno @ Serial 1148
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