Max-plus objects to study the complexity of graphs

关于一个离散的max-plus运输问题

On a discrete max-plus transportation problem

本文提出了解决热带最优运输问题的显式算法，指出在特定条件下最优热带计划可能不对应于完美匹配，并证明了解的唯一性罕见。

Max-plus Stochastic Control and Risk-sensitivity

In the Maslov idempotent probability calculus, expectations of random variables are defined so as to be linear with respect to max-plus addition and scalar multiplication. This paper considers control problems in which the objective is to minimize the max-plus expectation of some max-plus additive running cost. Such problems arise naturally as limits of some types of risk sensitive stochastic control problems. The value function is a viscosity solution to a quasivariational inequality (QVI) of dynamic programming. Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous Hamilton…

Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on…

Solutions of max-plus linear equations and large deviations

We generalise the Gartner-Ellis theorem of large deviations theory. Our results allow us to derive large deviation type results in stochastic optimal control from the convergence of generalised logarithmic moment generating functions. They rely on the characterisation of the uniqueness of the solutions of max-plus linear equations. We give an illustration for a simple investment model, in which logarithmic moment generating functions represent risk-sensitive values.

Cycle time of stochastic max-plus linear systems

We analyze the asymptotic behavior of sequences of random variables defined by an initial condition, a stationary and ergodic sequence of random matrices, and an induction formula involving multiplication is the so-called max-plus algebra. This type of recursive sequences are frequently used in applied probability as they model many systems as some queueing networks, train and computer networks, and production systems. We give a necessary condition for the recursive sequences to satisfy a strong law of large numbers, which proves to be sufficient when the matrices are i.i.d. Moreover, we const…

Law of Large Numbers for products of random matrices with coefficients in the max-plus semi-ring

We analyze the asymptotic behavior of random variables $x(n,x\_0)$ defined by $x(0,x\_0)=x\_0$ and $x(n+1,x\_0)=A(n)x(n,x\_0)$, where $\sAn$ is a stationary and ergodic sequence of random matrices with entries in the semi-ring \mbox{$\R\cup\{-\infty\}$} whose addition is the $\max$ and whose multiplication is $+$. Such sequences modelize a large class of discrete event systems, among which timed event graphs, 1-bounded Petri nets, some queuing networks, train or computer networks. We give necessary conditions for $(\frac{1}{n}x(n,x\_0))\_{n\in\N}$ to converge almost surely. Then, we prove a ge…