The max-plus finite element method for optimal control problems: further approximation results
Marianne Akian, Stephane Gaubert, Asma Lakhoua
备注：
13 pages, 2 figures
发表信息：
Proceedings of the joint 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005 (CDC-ECC'05), Sevilla, Spain, Dec. 2005, pages 4505 - 4510
We develop the max-plus finite element method to solve finite horizon deterministic optimal control problems. This method, that we introduced in a previous work, relies on a max-plus variational formulation, and exploits the properties of projectors on max-plus semimodules. We prove here a convergence result, in arbitrary dimension, showing that for a subclass of problems, the error estimate is of order $δ+Δx(δ)^{-1}$, where $δ$ and $Δx$ are the time and space steps respectively. We also show how the max-plus analogues of the mass and stiffness matrices can be computed by convex optimization, even when the global problem is non convex. We illustrate the method by numerical examples in dimension 2.
We establish the following max-plus analogue of Minkowski's theorem. Any point of a compact max-plus convex subset of $(R\cup\{-\infty\})^n$ can be written as the max-plus convex combination of at most $n+1$ of the extreme points of this subset. We establish related results for closed max-plus convex cones and closed unbounded max-plus convex sets. In particular, we show that a closed max-plus convex set can be decomposed as a max-plus sum of its recession cone and of the max-plus convex hull of its extreme points.
It is known that the solvability of a Sylvester equation over max-plus algebra can be determined in polynomial time by verifying its principal solution. A succinct representation of the principal solution is presented, with a more accurate computational complexity, for a Sylvester equation over max-plus algebra.
Recently, a max-plus dual space fundamental solution semigroup for a class of difference Riccati equation (DRE) has been developed. This fundamental solution semigroup is represented in terms of the kernel of a specific max-plus linear operator that plays the role of the dynamic programming evolution operator in a max-plus dual space. In order to fully understand connections between this dual space fundamental solution semigroup and evolution of the value function of the underlying optimal control problem, a new max-plus primal space fundamental solution semigroup for the same class of differe…
In this paper we show that the so-called array Fréchet problem in Probability/Statistics is (max, +)-linear. The upper bound of Fréchet is obtained using simple arguments from residuation theory and lattice distributivity. The lower bound is obtained as a loop invariant of a greedy algorithm. The algorithm is based on the max-plus linearity of the Fréchet problem and the Monge property of bivariate distribution.
A new fundamental solution semigroup for operator differential Riccati equations is developed. This fundamental solution semigroup is constructed via an auxiliary finite horizon optimal control problem whose value functional growth with respect to time horizon is determined by a particular solution of the operator differential Riccati equation of interest. By exploiting semiconvexity of this value functional, and the attendant max-plus linearity and semigroup properties of the associated dynamic programming evolution operator, a semigroup of max-plus integral operators is constructed in a dual…
We develop a max-plus spectral theory for infinite matrices. We introduce recurrence and tightness conditions, under which many results of the finite dimensional theory, concerning the representation of eigenvectors and the asymptotic behavior of powers of matrices, carry over. We also announce more general representation results for eigenvectors, which are obtained by introducing a max-plus analogue of the Martin boundary.
This note further addresses the global optimization problem for max-plus linear systems considered in [Automatica 119 (2020) 109104]. Firstly, the operations between infinity elemens and real numbers involved in the formulas of solving global optimization problems are explained explicitly. Secondly, the formula of the greatest lower bound and the criterion of solvability of globally optimal solutions are simplified. Thirdly, the criterion of uniqueness of globally optimal solutions and the set of all globally optimal solutions are presented.
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…
We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule S^n over a semiring S is rational if it has a generating family that is a rational subset of S^n, S^n being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules are stable under the natural algebraic operations (union, product, direct and inverse image, …
This paper deals with the state estimation for max-plus linear systems. This estimation is carried out following the ideas of the observer method for classical linear systems. The system matrices are assumed to be known, and the observation of the input and of the output is used to compute the estimated state. The observer design is based on the residuation theory which is suitable to deal with linear mapping inversion in idempotent semiring.
We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max-plus or tropical semiring. We establish a duality theorem relating both notions, which we use to construct dynamic observers. These are useful in situations in which some of the system coefficients may vary within certain intervals. The results are illustrated by an application to a manufacturing system.
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 i…
Efficient Riccati equation based techniques for the approximate solution of discrete time linear regulator problems are restricted in their application to problems with quadratic terminal payoffs. Where non-quadratic terminal payoffs are required, these techniques fail due to the attendant non-quadratic value functions involved. In order to compute these non-quadratic value functions, it is often necessary to appeal directly to dynamic programming in the form of grid- or element-based iterations for the value function. These iterations suffer from poor scalability with respect to problem dimen…
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.
We consider the problem of certifying an inequality of the form $f(x)\geq 0$, $\forall x\in K$, where $f$ is a multivariate transcendental function, and $K$ is a compact semialgebraic set. We introduce a certification method, combining semialgebraic optimization and max-plus approximation. We assume that $f$ is given by a syntaxic tree, the constituents of which involve semialgebraic operations as well as some transcendental functions like $\cos$, $\sin$, $\exp$, etc. We bound some of these constituents by suprema or infima of quadratic forms (max-plus approximation method, initially introduce…
The concept of (A,B)-invariant subspace (or controlled invariant) of a linear dynamical system is extended to linear systems over the max-plus semiring. Although this extension presents several difficulties, which are similar to those encountered in the same kind of extension to linear dynamical systems over rings, it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings. Sufficient conditions are given for computing the maximal (A,B)-invariant subspace contained in a given space and the existence of linear state feedbacks is d…
National Technical University of Athens（雅典国家技术大学）；University of Pennsylvania（宾夕法尼亚大学）
In this work, we study the problem of finding approximate, with minimum support set, solutions to matrix max-plus equations, which we call sparse approximate solutions. We show how one can obtain such solutions efficiently and in polynomial time for any $\ell_p$ approximation error. Based on these results, we propose a novel method for piecewise-linear fitting of convex multivariate functions, with optimality guarantees for the model parameters and an approximately minimum number of affine regions.
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 a…
We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be equivalent to the minimum set cover problem and, thus, NP-complete. In the latter one, the approximation is quantified by the $\ell_{1}$ residual error norm, which is shown to have supermodular properties under some convex constraints, called lateness constraints. Thus, greedy approximation algorithms of polynomial complexity can be employed for both problems with g…
This thesis proposes mathematical traffic models and control laws for metro lines with one junction. The models are based on the ones developed for linear metro lines (without junction) in [12, 14]. The train dynamics are described with a discrete event traffic model. Two time constraints are considered. The first one imposes lower bounds on the train run and dwell times. The second one fixes a lower bound on the safe separation time between two trains. A model of the train dynamics on the junction is proposed, as well as control laws for the train run and dwell times, as a function of the pas…
A robust (deterministic) filtering approach to the problem of optimal sensor selection is considered herein. For a given system with several sensors, at each time step the output of one of the sensors must be chosen in order to obtain the best state estimate. We reformulate this problem in an optimal control framework which can then be solved using dynamic programming. In order to tackle the numerical computation of the solution in an efficient manner, we exploit the preservation of the min-plus structure of the optimal cost function when acted upon by the dynamic programming operator. This te…
A general problem in optimal control consists of finding a terminal reward that makes the value function independent of the horizon. Such a terminal reward can be interpreted as a max-plus eigenvector of the associated Lax-Oleinik semigroup. We give a representation formula for all these eigenvectors, which applies to optimal control problems in which the state space is non compact. This representation involves an abstract boundary of the state space, which extends the boundary of metric spaces defined in terms of Busemann functions (the horoboundary). Extremal generators of the eigenspace cor…
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 algor…
Max-plus based methods have been recently explored for solution of first-order Hamilton-Jacobi-Bellman equations by several authors. In particular, McEneaney's curse-of-dimensionality free method applies to the equations where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. In previous works of McEneaney and Kluberg, the approximation error of the method was shown to be $O(1/(Nτ))+O(\sqrtτ)$ where $τ$ is the time discretization step and $N$ is the number of iterations. Here we use a recently established contraction result for the indefinite Riccati flow in Th…
Using the tools of optimal control, semiconvex duality and \maxp algebra, this work derives a unifying representation of the solution for the matrix differential Riccati equation (DRE) with time-varying coefficients. It is based upon a special case of the \maxp fundamental solution, first proposed in \cite{FlemMac}. Such fundamental solution can extend a special solution of certain bivariate DRE into the general solution, and can analytically solve the DRE starting from any initial condition. This paper also shows that under a fixed duality kernel, the semiconvex dual of a DRE solution satisfi…
The research problem in this work is the relaxation of maximizing non-negative submodular plus modular with the entire real number domain as its value range over a family of down-closed sets. We seek a feasible point $\mathbf{x}^*$ in the polytope of the given constraint such that $\mathbf{x}^*\in\arg\max_{\mathbf{x}\in\mathcal{P}\subseteq[0,1]^n}F(\mathbf{x})+L(\mathbf{x})$, where $F$, $L$ denote the extensions of the underlying submodular function $f$ and modular function $\ell$. We provide an approximation algorithm named \textsc{Measured Continuous Greedy with Adaptive Weights}, which yiel…
We show that the set of max-plus measures representing a given max-plus harmonic vector has a least element. This may be viewed as an analogue of the uniqueness of the integral representation of harmonic functions in Potential Theory. As an application, we show how the distance-like functions of a metric space can be expressed in terms of the Busemann points of the metric boundary.
