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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 14 (9 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 Minkowski
Convex Analysis
, 1970
"... In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a lo ..."
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Cited by 5350 (67 self)
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In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a long time, ‘variational ’ problems have been identified mostly with the ‘calculus of variations’. In that venerable subject, built around the minimization of integral functionals, constraints were relatively simple and much of the focus was on infinitedimensional function spaces. A major theme was the exploration of variations around a point, within the bounds imposed by the constraints, in order to help characterize solutions and portray them in terms of ‘variational principles’. Notions of perturbation, approximation and even generalized differentiability were extensively investigated. Variational theory progressed also to the study of socalled stationary points, critical points, and other indications of singularity that a point might have relative to its neighbors, especially in association with existence theorems for differential equations.
The Minkowski Theorem for Maxplus Convex Sets
, 2006
"... We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones a ..."
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Cited by 36 (14 self)
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We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones
The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
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Cited by 711 (0 self)
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algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm
Two Lectures on Maxplus Algebra
 In Proceedings of the 26th Spring School on Theoretical Computer Science and Automatic Control, Noirmoutier
, 1998
"... 5 1. Introduction: the (max, +) and tropical semirings 5 2. Seven good reasons to use the (max, +) semiring 6 3. Solving Linear Equations in the (max, +) Semiring 11 Chapter 2. Exotic Semirings: Examples and General Results 21 1. Definitions and Zoology 21 2. Combinatorial Formul in Semirings ..."
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Cited by 3 (1 self)
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25 3. Naturally Ordered Semirings 32 4. Semimodules and Linear Maps 34 5. Images and Kernels 36 6. Factorization of Linear Maps and Linear Extension Theorem 37 7. Finiteness Theorems for Semimodules 39 8. Minimal Generating Families, Convex Geometries, and MaxPlus Projective Geometry 41 9
Establishing a Metric in MaxPlus Geometry
"... Abstract. Using the characterization of the segments in the maxplus semimodule R n max, provided by Nitica and Singer in [5], we find a class of metrics on the finite part of R n max. One of them is the Euclidean length of the maxplus segment connecting two points. This metric is not quasiconvex. ..."
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Abstract. Using the characterization of the segments in the maxplus semimodule R n max, provided by Nitica and Singer in [5], we find a class of metrics on the finite part of R n max. One of them is the Euclidean length of the maxplus segment connecting two points. This metric is not quasiconvex
MULTIORDER, KLEENE STARS AND CYCLIC PROJECTORS IN THE GEOMETRY OF MAX CONES
, 2008
"... 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 finitedimension ..."
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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
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 484 (3 self)
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The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological
Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise
, 2006
"... This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that ..."
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Cited by 496 (2 self)
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. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure
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