Tolerance types of interval eigenvectors in max-plus algebra

Gavalec, Martin; Plávka, Ján; Ponce, Daniela

doi:10.1016/j.ins.2016.05.023

Cited by 18 publications

(7 citation statements)

References 15 publications

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“…The interval eigenproblem for A and X consists of recognizing whether A ⊗ x = λ ⊗ x holds true for A ∈ A, x ∈ X, λ ∈ B, with suitable quantifiers (e.g., for all A ∈ A, for some A ∈ A, for all x ∈ X, for some x ∈ X) and their various combinations. According to the choice of quantifiers and their order, several different types of interval eigenvectors can be defined (see, e.g., [11][12][13] for similar classification). The following three types are studied in detail in this paper.…”

Section: Remarkmentioning

confidence: 99%

“…This section uses the general properties of the eigenvalue in interval eigenproblems that have been demonstrated in [11]. Suppose that there is an interval matrix A = [A, A] and a vector x ∈ B(n).…”

Section: Weak Interval Eigenvectors In a Max-plus Algebramentioning

confidence: 99%

“…If the conditions (26) and (27) are satisfied for given λ ∈ B and x ∈ X, then the matrix A with A ⊗ x = λ ⊗ x can be constructed by the following procedure, described in [11], Lemma 4.2. In view of the condition…”

Section: Remarkmentioning

confidence: 99%

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Strong, Strongly Universal and Weak Interval Eigenvectors in Max-Plus Algebra

2020

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The optimization problems, such as scheduling or project management, in which the objective function depends on the operations maximum and plus, can be naturally formulated and solved in max-plus algebra. A system of discrete events, e.g., activations of processors in parallel computing, or activations of some other cooperating machines, is described by a systems of max-plus linear equations. In particular, if the system is in a steady state, such as a synchronized computer network in data processing, then the state vector is an eigenvector of the system. In reality, the entries of matrices and vectors are considered as intervals. The properties and recognition algorithms for several types of interval eigenvectors are studied in this paper. For a given interval matrix and interval vector, a set of generators is defined. Then, the strong and the strongly universal eigenvectors are studied and described as max-plus linear combinations of generators. Moreover, a polynomial recognition algorithm is suggested and its correctness is proved. Similar results are presented for the weak eigenvectors. The results are illustrated by numerical examples. The results have a general character and can be applied in every max-plus algebra and every instance of the interval eigenproblem.

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Section: Remarkmentioning

confidence: 99%

Section: Weak Interval Eigenvectors In a Max-plus Algebramentioning

confidence: 99%

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Strong, Strongly Universal and Weak Interval Eigenvectors in Max-Plus Algebra

2020

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“…The max-plus algebra has appeared under the name extremal algebra for many years [ 1 , 2 ]. In this algebra, several types of solvability of interval linear systems, including both interval linear equations and interval linear inequalities, have been studied in the literature, see, e.g., weak and strong solvability in [ 3 , 4 ], tolerance solvability in [ 5 – 7 ], and control solvability in [ 8 ]. A detailed discussion of interval solutions can be found in Chap.…”

Section: Introductionmentioning

confidence: 99%

AE solutions to two-sided interval linear systems over max-plus algebra

Wang

2018

J Inequal Appl

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This paper introduces a concept of AE solutions to two-sided interval max-plus linear systems, a rather general concept which includes many known concepts of solutions to interval systems, in particular, weak, strong, tolerance and control solutions as its special cases. We state full characterizations of AE solutions for the two-sided interval max-plus systems, including both linear inequalities and linear equations. Moreover, we provide a specific example to illustrate an efficient method of finding the AE solution set.

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“…In classical linear algebra this leads to the notions of united solutions, controllable solutions and tolerable solutions [17,20]. In max-algebra we similarly have, e.g., four types of interval extensions of the max-algebraic spectral problem [8] or two types of interval extensions of robustness studied in [15].…”

Section: Introductionmentioning

confidence: 99%

Reachability of eigenspaces for interval circulant matrices in max-algebra

Plávka

Sergeev

2018

Linear Algebra and its Applications

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A nonnegative matrix A is said to be strongly robust if its max-algebraic eigencone is universally reachable, i.e., if the orbit of any initial vector ends up with a max-algebraic eigenvector of A. Consider the case when the initial vector is restricted to an interval and A can be any matrix from a given interval of nonnegative circulant matrices. The main aim of this paper is to classify and characterize the six types of interval robustness in this situation. This naturally leads us also to study the max-algebraic spectral theory of circulant matrices and the relation of inclusion between attraction cones of circulant matrices in max-algebra.

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Tolerance types of interval eigenvectors in max-plus algebra

Cited by 18 publications

References 15 publications

Strong, Strongly Universal and Weak Interval Eigenvectors in Max-Plus Algebra

Strong, Strongly Universal and Weak Interval Eigenvectors in Max-Plus Algebra

AE solutions to two-sided interval linear systems over max-plus algebra

Reachability of eigenspaces for interval circulant matrices in max-algebra

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