Max-plus Algebraic Tools for Discrete Event Systems, Static Analysis, and Zero-Sum Games

Gaubert, Stéphane

doi:10.1007/978-3-642-04368-0_2

Cited by 1 publication

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“…Other applications include the max-algebraic approach to optimal control (Litvinov et al, 2001;McEneaney, 2006), general max-plus dynamical systems and control (Adzkiya et al, 2015;Hardouin et al, 2011) and generalized HMMs for audiovisual event detection (Maragos and Koutras, 2015). An extensive survey about the applications of the max-plus algebra can be found in Gaubert (2009). Generalizations of max-plus algebra using other idempotent semirings are described in Gondran and Minoux and the latter problem respectively.…”

Section: Introductionmentioning

confidence: 99%

Sparsity in max-plus algebra and systems

Tsiamis

Maragos

2019

Discrete Event Dyn Syst

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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 ℓ 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 guaranteed bounds of approximation. We also study the sparse recovery problem and present conditions, under which, the sparsest exact solution solves it. Through multi-machine interactive processes, we describe how the present framework could be applied to two practical discrete event systems problems: resource optimization and structure-seeking system identification. We also show how sparsity is related to the pruning problem. Finally, we present a numerical example of the structure-seeking system identification problem and we study the performance of the greedy algorithm via simulations. * The paper was published in the Discrete Event (2008). A unification of max-type algebras and their duals using weighted lattices with applications to nonlinear dynamical systems was presented in Maragos (2017). Meanwhile, in the last decade, we have experienced an increase of interest in sparsity in linear equations and linear systems. A solution of a linear equation is sparse when it has many zero elements. The reason we are interested in such solutions, is that they need less elements to describe the same information. They provide us a way of compressing the available data, throwing away those that are unnecessary (Donoho, 2006). They also reveal the structure of partially known signals (Candès et al, 2006) or systems (Chen et al, 2009). In control systems, sparsity has been sought in the sense of minimizing the number of sensors or actuators, subject to energy (Tzoumas et al, 2016;Summers et al, 2016) or observability-controllability constraints (Pequito et al, 2016).Although sparsity has been extensively studied in the linear setting (Elad, 2010), it is still not much developed in more general nonlinear settings. In this work, we aim to define and study sparsity in the max-plus algebraic setting. A sparse solution of a max-plus equation is a solution with many non-informative elements, i.e. the infinite elements. As in the linear case, such solutions use the least number of elements to describe the same information, thus yielding compressed data. But there are many other applications where sparsity could be relevant. For example, in max-plus systems (either static or dynamical), finding sparse inputs implies that we are activating fewer actuators/machines, thus, saving resources. Similarly, the problem of selecting few sensors to observe a max-plus system could be expressed in terms of designing sparse output matrices. Another application co...

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