Max-plus convex sets and max-plus semispaces. I

Niţică, Viorel; Singer, Ivan

doi:10.1080/02331930600819852

Cited by 22 publications

(40 citation statements)

References 6 publications

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“…For points in the elementary segment, the coordinates in the first subset are constant and the coordinates in the second subset are all equal to a parameter running over a 1-dimensional interval. Therefore, similarly to the max-plus case (see [21], Remark 4.3) in B d there are elementary segments in only 2 d − 1 directions. Elementary segments are the "building blocks" for the max-min segments in B d , in the sense that every segment [x, y] ⊆ B d is the concatenation of a finite number of elementary subsegments (at most) 2d − 1, respectively 2d − 2, in the case of comparable, respectively incomparable, endpoints.…”

Section: Max-min Segmentsmentioning

confidence: 77%

On the dimension of max–min convex sets

Niţică

Sergeev

2015

Fuzzy Sets and Systems

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We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property.

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Section: Max-min Segmentsmentioning

confidence: 77%

On the dimension of max–min convex sets

Niţică

Sergeev

2015

Fuzzy Sets and Systems

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“…In this note we give some complements to our article [13]. In section 2 we show that the theories of max-plus convexity in R n max ¼ ðR n max , , þÞ in the sense of Zimmermann ( [19,5]) and B-convexity in R n þ ¼ ðR n þ , max , ÂÞ in the sense of Briec and Horvath ( [3,4]) are equivalent, via the well-known isomorphism between these structures, and we deduce some consequences.…”

Section: Introductionmentioning

confidence: 91%

“…In the paper [13], we have studied ''max-plus convex sets'' and ''max-plus semispaces'' in the semimodule R n max over the max-plus semifield R max : These are important, among others, in understanding the ''max-linear'' structure of R n max (see [13] and, for some recent advances, [14]). …”

Section: Introductionmentioning

confidence: 99%

Max-plus convex sets and max-plus semispaces. II

Niţică¹,

Singer²

2007

Optimization

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We give some complements to Nitica, V. and Singer, I., 2007, Max-plus convex sets and max-plus semispaces, I. Optimization, 56, 171-205. We show that the theories of max-plus convexity in R n max and B-convexity in R n þ are equivalent, and we deduce some consequences. We show that max-plus convexity in R n is a multi-order convexity. We give simpler proofs, using only the definition of max-plus segments, of the results of loc. cit. on max-plus semispaces. We show that unless is a total order on A, the results of loc. cit. on semispaces cannot be generalized in a natural way to the framework of A n ¼ ðA n , ; Þ, where A :¼ M [ fÀ1g, with M ¼ ðM, ; Þ being a lattice ordered group and À1 a ''least element'' adjoined to M.

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“…For completeness of the presentation here, we recall some results from the theory of min-plus convexity [4,8,12,20,23,24]. In order to proceed more quickly to the delay game model, where proofs are needed, they have been moved to the appendix.…”

Section: Min-plus Convexitymentioning

confidence: 99%

“…It may be helpful to refer to Figure 8, which provides a high-level flow chart of this backward propagation. Suppose one has constructed a terminal payoff function,ψ, in the form of an FCMPCF as in (20). Without loss of generality, we have the value at the terminal time given, as in (20) - (21), by …”

Section: Comments On Computationmentioning

confidence: 99%