arXiv subjects

Marianne Akian

Publications and source records attributed to Marianne Akian.

At least 37 records · Page 2Linked to original sources

arXiv2020

Tropical Dynamic Programming for Lipschitz Multistage Stochastic Programming

We present an algorithm called Tropical Dynamic Programming (TDP) which builds upper and lower approximations of the Bellman value functions in risk-neutral Multistage Stochastic Programming (MSP), with independent noises of finite supports. To tackle the curse of dimensionality, popular parametric variants of Approximate Dynamic Programming approximate the Bellman value function as linear combinations of basis functions. Here, Tropical Dynamic Programming builds upper (resp. lower) approximations of a given value function as min-plus linear (resp. max-plus linear) combinations of "basic functions". At each iteration, TDP adds a new basic function to the current combination following a deterministic criterion introduced by Baucke, Downward and Zackeri in 2018 for a variant of Stochastic Dual Dynamic Programming. We prove, for every Lipschitz MSP, the asymptotic convergence of the generated approximating functions of TDP to the Bellman value functions on sets of interest. We illustrate this result on MSP with linear dynamics and polyhedral costs.

math.OC↗

arXiv2020

A stochastic algorithm for deterministic multistage optimization problems

Several attempts to dampen the curse of dimensionnality problem of the Dynamic Programming approach for solving multistage optimization problems have been investigated. One popular way to address this issue is the Stochastic Dual Dynamic Programming method (SDDP) introduced by Perreira and Pinto in 1991 for Markov Decision Processes. Assuming that the value function is convex (for a minimization problem), one builds a non-decreasing sequence of lower (or outer) convex approximations of the value function. Those convex approximations are constructed as a supremum of affine cuts. On continuous time deterministic optimal control problems, assuming that the value function is semiconvex, Zheng Qu, inspired by the work of McEneaney, introduced in 2013 a stochastic max-plus scheme that builds upper (or inner) non-increasing approximations of the value function. In this note, we build a common framework for both the SDDP and a discrete time version of Zheng Qu's algorithm to solve deterministic multistage optimization problems. Our algorithm generates monotone approximations of the value functions as a pointwise supremum, or infimum, of basic (affine or quadratic for example) functions which are randomly selected. We give sufficient conditions on the way basic functions are selected in order to ensure almost sure convergence of the approximations to the value function on a set of interest.

math.OC↗

arXiv2020

Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area

We portray the evolution of the Covid-19 epidemic during the crisis of March-April 2020 in the Paris area, by analyzing the medical emergency calls received by the EMS of the four central departments of this area (Centre 15 of SAMU 75, 92, 93 and 94). Our study reveals strong dissimilarities between these departments. We show that the logarithm of each epidemic observable can be approximated by a piecewise linear function of time. This allows us to distinguish the different phases of the epidemic, and to identify the delay between sanitary measures and their influence on the load of EMS. This also leads to an algorithm, allowing one to detect epidemic resurgences. We rely on a transport PDE epidemiological model, and we use methods from Perron-Frobenius theory and tropical geometry.

stat.AP↗

arXiv2020

A convex programming approach to solve posynomial systems

We exhibit a class of classical or tropical posynomial systems which can be solved by reduction to linear or convex programming problems. This relies on a notion of colorful vectors with respect to a collection of Newton polytopes. This extends the convex programming approach of one player stochastic games.

math.OC↗

arXiv2019

Solving Ergodic Markov Decision Processes and Perfect Information Zero-sum Stochastic Games by Variance Reduced Deflated Value Iteration

Recently, Sidford, Wang, Wu and Ye (2018) developed an algorithm combining variance reduction techniques with value iteration to solve discounted Markov decision processes. This algorithm has a sublinear complexity when the discount factor is fixed. Here, we extend this approach to mean-payoff problems, including both Markov decision processes and perfect information zero-sum stochastic games. We obtain sublinear complexity bounds, assuming there is a distinguished state which is accessible from all initial states and for all policies. Our method is based on a reduction from the mean payoff problem to the discounted problem by a Doob h-transform, combined with a deflation technique. The complexity analysis of this algorithm uses at the same time the techniques developed by Sidford et al. in the discounted case and non-linear spectral theory techniques (Collatz-Wielandt characterization of the eigenvalue).

math.OC↗

arXiv2019

A bilevel optimization model for load balancing in mobile networks through price incentives

We propose a model of incentives for data pricing in large mobile networks, in which an operator wishes to balance the number of connections (active users) of different classes of users in the different cells and at different time instants, in order to ensure them a sufficient quality of service. We assume that each user has a given total demand per day for different types of applications, which he may assign to different time slots and locations, depending on his own mobility, on his preferences and on price discounts proposed by the operator. We show that this can be cast as a bilevel programming problem with a special structure allowing us to develop a polynomial time decomposition algorithm suitable for large networks. First, we determine the optimal number of connections (which maximizes a measure of balance); next, we solve an inverse problem and determine the prices generating this traffic. Our results exploit a recently developed application of tropical geometry methods to mixed auction problems, as well as algorithms in discrete convexity (minimization of discrete convex functions in the sense of Murota). We finally present an application on real data provided by Orange and we show the efficiency of the model to reduce the peaks of congestion.

math.OC↗

arXiv2018

A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors

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 $\mathbb{R}_{\scriptscriptstyle >0}^n$. 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 $α, β> 0$, the "slice space" $\mathcal{S}_α^β:= \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \mid αx \leq f(x) \leq βx \}$ 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.

math.OC↗

arXiv2018

Log-sum-exp neural networks and posynomial models for convex and log-log-convex data

We show in this paper that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is an universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named LSET. Under a suitable exponential transformation, the class of LSET functions maps to a family of generalized posynomials GPOST, which we similarly show to be universal approximators for log-log-convex functions. A key feature of an LSET network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.

cs.NE↗

arXiv2018

Tropical compound matrix identities

We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and~$\adj(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well.

math.AC↗

arXiv2018

From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton-Jacobi-Bellman equations

In a previous work (Akian, Fodjo, 2016), we introduced a lower complexity probabilistic max-plus numerical method for solving fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. This method was based on the idempotent expansion properties obtained by McEneaney, Kaise and Han (2011) and on the numerical probabilistic method proposed by Fahim, Touzi and Warin (2011) for solving some fully nonlinear parabolic partial differential equations. A difficulty of the algorithm of Fahim, Touzi and Warin is in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence of the algorithm. Here, we propose a new "probabilistic scheme" which is monotone under rather weak assumptions, including the case of strongly elliptic PDE with bounded coefficients. This allows us to apply our probabilistic max-plus method in more general situations. We illustrate this on the evaluation of the superhedging price of an option under uncertain correlation model with several underlying stocks and changing sign cross gamma, and consider in particular the case of 5 stocks leading to a PDE in dimension 5.

math.OC↗

arXiv2018

Probabilistic max-plus schemes for solving Hamilton-Jacobi-Bellman equations

We consider fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. In previous works (Akian, Fodjo, 2016 and 2017), we introduced a lower complexity probabilistic numerical algorithm for such equations by combining max-plus and numerical probabilistic approaches. The max-plus approach is in the spirit of the one of McEneaney, Kaise and Han (2011), and is based on the distributivity of monotone operators with respect to suprema. The numerical probabilistic approach is in the spirit of the one proposed by Fahim, Touzi and Warin (2011). A difficulty of the latter algorithm was in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence of the algorithm. Here, we present new probabilistic schemes which are monotone under rather weak assumptions, and show error estimates for these schemes. These estimates will be used in further works to study the probabilistic max-plus method.

math.OC↗

arXiv2017

Minimax representation of nonexpansive functions and application to zero-sum recursive games

We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on $L^\infty$ spaces and on $\mathbb{R}^n$, which extend results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representation and polyhedral approximation for the class of Shapley operators arising from games without instantaneous payments (Everett's recursive games).

math.OC↗

arXiv2017

Generic uniqueness of the bias vector of finite stochastic games with perfect information

Mean-payoff zero-sum stochastic games can be studied by means of a nonlinear spectral problem. When the state space is finite, the latter consists in finding an eigenpair $(u,λ)$ solution of $T(u)=λe + u$, where $T:\mathbb{R}^n \to \mathbb{R}^n$ is the Shapley (or dynamic programming) operator, $λ$ is a scalar, $e$ is the unit vector, and $u \in \mathbb{R}^n$. The scalar $λ$ yields the mean payoff per time unit, and the vector $u$, called the bias, allows one to determine optimal stationary strategies. The existence of the eigenpair $(u,λ)$ is generally related to ergodicity conditions. A basic issue is to understand for which classes of games the bias vector is unique (up to an additive constant). In this paper, we consider perfect-information zero-sum stochastic games with finite state and action spaces, thinking of the transition payments as variable parameters, transition probabilities being fixed. We show that the bias vector, thought of as a function of the transition payments, is generically unique (up to an additive constant). The proof uses techniques of max-plus (or tropical) algebra and nonlinear Perron-Frobenius theory. As an application of our results, we obtain a perturbation scheme allowing one to solve degenerate instances of stochastic games by policy iteration.

math.OC↗

arXiv2016

Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and Pólya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated.

math.SP↗

arXiv2016

A probabilistic max-plus numerical method for solving stochastic control problems

We consider fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. We construct a lower complexity probabilistic numerical algorithm by combining the idempotent expansion properties obtained by McEneaney, Kaise and Han (2011) for solving such problems with a numerical probabilistic method such as the one proposed by Fahim, Touzi and Warin (2011) for solving some fully nonlinear parabolic partial differential equations. Numerical tests on a small example of pricing and hedging an option are presented.

math.OC↗

arXiv2016

Non-archimedean valuations of eigenvalues of matrix polynomials

We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials.

math.SP↗

arXiv2015

Hypergraph conditions for the solvability of the ergodic equation for zero-sum games

The ergodic equation is a basic tool in the study of mean-payoff stochastic games. Its solvability entails that the mean payoff is independent of the initial state. Moreover, optimal stationary strategies are readily obtained from its solution. In this paper, we give a general sufficient condition for the solvability of the ergodic equation, for a game with finite state space but arbitrary action spaces. This condition involves a pair of directed hypergraphs depending only on the ``growth at infinity'' of the Shapley operator of the game. This refines a recent result of the authors which only applied to games with bounded payments, as well as earlier nonlinear fixed point results for order preserving maps, involving graph conditions.

math.OC↗

arXiv2015

Ergodicity conditions for zero-sum games

A basic question for zero-sum repeated games consists in determining whether the mean payoff per time unit is independent of the initial state. In the special case of "zero-player" games, i.e., of Markov chains equipped with additive functionals, the answer is provided by the mean ergodic theorem. We generalize this result to repeated games. We show that the mean payoff is independent of the initial state for all state-dependent perturbations of the rewards if and only if an ergodicity condition is verified. The latter is characterized by the uniqueness modulo constants of nonlinear harmonic functions (fixed points of the recession function associated to the Shapley operator), or, in the special case of stochastic games with finite action spaces and perfect information, by a reachability condition involving conjugate subsets of states in directed hypergraphs. We show that the ergodicity condition for games only depends on the support of the transition probability, and that it can be checked in polynomial time when the number of states is fixed.

math.OC↗