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Generators, Extremals and Bases of Max Cones
, 2006
"... We give simple algebraic proofs of results on generators and bases of max cones, some of which are known. We show that every generating set S for a cone in max algebra can be partitioned into two parts: the independent set of extremals E in the cone and a set F every member of which is redundant in ..."
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Cited by 15 (6 self)
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We give simple algebraic proofs of results on generators and bases of max cones, some of which are known. We show that every generating set S for a cone in max algebra can be partitioned into two parts: the independent set of extremals E in the cone and a set F every member of which is redundant in S. We exploit the result that extremals are minimal elements under suitable scalings of vectors. We also give an algorithm for finding the (essentially unique) basis of a finitely generated cone.
Tropical halfspaces
"... Abstract. As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R, min, +). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels [2004] in a variety of contexts. The key tool to the ..."
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Cited by 13 (0 self)
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Abstract. As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R, min, +). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels [2004] in a variety of contexts. The key tool to the understanding is a newly defined sign of the tropical determinant, which shares remarkably many properties with the ordinary sign of the determinant of a matrix. The methods are used to obtain an optimal tropical convex hull algorithm in two dimensions. 1.
Maxplus convex geometry
 of Lecture Notes in Comput. Sci
, 2006
"... Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theore ..."
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Cited by 8 (6 self)
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Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theorem, and the characterization of the analogues of “simplicial ” cones in terms of distributive lattices. 1
Solutions of maxplus linear equations and large deviations
 in "Proceedings of the joint 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005 (CDCECC’05), Seville, Espagne", Also arXiv:math.PR/0509279, 2005, http://hal.inria.fr/inria00000218/en/. Maxplus 37
"... Abstract — We generalise the GärtnerEllis 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 ..."
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Cited by 7 (1 self)
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Abstract — We generalise the GärtnerEllis 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 maxplus linear equations. We give an illustration for a simple investment model, in which logarithmic moment generating functions represent risksensitive values. I.
Minimal halfspaces and external representation of tropical polyhedra
, 2009
"... We give a characterization of the minimal tropical halfspaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal halfspaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving ..."
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Cited by 3 (3 self)
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We give a characterization of the minimal tropical halfspaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal halfspaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the MinkowskiWeyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of halfspaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.
KERNEL THEOREMS AND NUCLEARITY IN IDEMPOTENT MATHEMATICS. AN ALGEBRAIC APPROACH
, 2006
"... In the framework of idempotent mathematics, analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck are studied. Idempotent versions of nuclear spaces (in the sense of A. Grothendieck) are discussed. The socalled algebraic approach is used. This means that the basic concepts and ..."
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Cited by 1 (0 self)
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In the framework of idempotent mathematics, analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck are studied. Idempotent versions of nuclear spaces (in the sense of A. Grothendieck) are discussed. The socalled algebraic approach is used. This means that the basic concepts and results (including those of “topological” nature) are simulated in purely algebraic terms.
MULTIORDER, KLEENE STARS AND CYCLIC PROJECTORS IN THE GEOMETRY OF MAX CONES
, 807
"... Abstract. This paper summarizes results on some topics in the maxplus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to maxplus analogues of some statements in the finite ..."
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Abstract. This paper summarizes results on some topics in the maxplus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to maxplus analogues of some statements in the finitedimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are fundamental for max algebra, as they accumulate the weights of optimal paths and describe the eigenspace of a matrix. On the other hand, the approach of tropical convexity decomposes a finitely generated semimodule into a number of convex regions, and these regions are column spans of uniquely defined Kleene stars. Another recent geometric result, that several semimodules with zero intersection can be separated from each other by maxplus halfspaces, leads to investigation of specific nonlinear operators called cyclic projectors. These nonlinear operators can be used to find a solution to homogeneous multisided systems of maxlinear equations. The results are presented in the setting of max cones, i.e., semimodules over the maxtimes semiring. 1.
THEME Modeling, Optimization, and Control of Dynamic SystemsTable of contents
"... 4.3. Switched systems 5 5. Software................................................................................. 6 6. New Results.............................................................................. 6 6.1. New results: geometric control 6 6.2. New results: quantum control 8 6.3. New res ..."
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4.3. Switched systems 5 5. Software................................................................................. 6 6. New Results.............................................................................. 6 6.1. New results: geometric control 6 6.2. New results: quantum control 8 6.3. New results: neurophysiology 8 6.4. New results: switched systems 9