Tropicalizing the Simplex Algorithm

Allamigeon, Xavier; Benchimol, Pascal; Gaubert, Stéphane; Joswig, Michael

doi:10.1137/130936464

Cited by 47 publications

(122 citation statements)

References 40 publications

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“…A proper intersection of multiplicity 4 can (up to symmetry) only occur on an edge of slope (4,1). A line intersecting the open interior of such an edge cannot lift to a bitangent by Theorem 3.1.…”

Section: 2mentioning

confidence: 95%

Lifting tropical bitangents

Len

Markwig

2020

Journal of Symbolic Computation

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We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents; however the conditions to determine whether a tropical bitangent has real lifts are not purely combinatorial.2010 Mathematics Subject Classification. 14T05.

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Section: 2mentioning

confidence: 95%

Lifting tropical bitangents

Len

Markwig

2020

Journal of Symbolic Computation

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“…One of them is how to find a point of a finite min-essential set S min (TP 2 ) that attains min y ′ {(x k ) T y ′ : y ′ ∈ TP 2 } and which min-essential set to choose. An option here is to exploit the tropical simplex method of Allamigeon, Benchimol, Gaubert and Joswig [1], which (under some generically true conditions imposed on TP 2 ) can find a point that attains min y ′ {(x k ) T y ′ : y ′ ∈ TP 2 } and belongs to the set of tropical basic points of TP 2 . The set of tropical basic points is finite and includes all extreme points [1] and hence all the minimal points of TP 2 , thus it is also a finite min-essential subset of TP 2 by Lemma 2.1.…”

Section: The Min-min and Max-min Problemmentioning

confidence: 99%

“…Step When f (x, y) = a T x ⊕ b T y, this procedure reduces the problem to a finite number of tropical linear programming problems solved, e.g., by the algorithms of [1,3,7]. Example 2.1 Consider the following numerical example in two-dimensional case.…”

Section: Proposition 23mentioning

confidence: 99%

“…However, these solutions are always dominated by the optimal points of partition 1 and partition 2. Therefore, in this example, it is sufficient to consider only partition 1 and partition 2. and decide between (x, y) 1 = ((−2, −3), (2, −1)) and (x, y) 2 = ((−3, −1), (1,1). Taking a 1 = a 2 = b 1 = b 2 makes (x, y) 2 an optimal solution of the problem, but taking a 2 = 10 and a 1 = b 1 = b 2 results in (x, y) 1 .…”

Section: Proposition 23mentioning

confidence: 99%

“…Our goal is to study some analogues and generalisations of problem (1) over the tropical (max-plus) semiring. This is a special case of a general idea to develop tropical bilevel optimization, inspired both by the well-developed methodology of bilevel optimization and some of the recent successes in tropical convexity and tropical optimization [1,2,7].…”

Section: Introductionmentioning

confidence: 99%

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Tropical Analogues of a Dempe-Franke Bilevel Optimization Problem

Sergeev

Liu

2019

Advances in Intelligent Systems and Computing

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We consider the tropical analogues of a particular bilevel optimization problem studied by Dempe and Franke [4] and suggest some methods of solving these new tropical bilevel optimization problems. In particular, it is found that the algorithm developed by Dempe and Franke can be formulated and its validity can be proved in a more general setting, which includes the tropical bilevel optimization problems in question. We also show how the feasible set can be decomposed into a finite number of tropical polyhedra, to which the tropical linear programming solvers can be applied.

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Solving Mean-Payoff Games via Quasi Dominions

Benerecetti

Dell’Erba

Mogavero

2020

Lecture Notes in Computer Science

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We propose a novel algorithm for the solution of mean-payoff games that merges together two seemingly unrelated concepts introduced in the context of parity games, small progress measures and quasi dominions. We show that the integration of the two notions can be highly beneficial and significantly speeds up convergence to the problem solution. Experiments show that the resulting algorithm performs orders of magnitude better than the asymptotically-best solution algorithm currently known, without sacrificing on the worst-case complexity.A two-player turn-based arena is a tuple A = Ps ⊕ , Ps ⊟ , Mv , with Ps ⊕ ∩ Ps ⊟ = ∅ and Ps Ps ⊕ ∪ Ps ⊟ , such that Ps, Mv is a finite directed graph without sinks. Ps ⊕ (resp., Ps ⊟ ) is the set of 1. The experiments were carried out on a 64-bit 3.9GHz quad-core machine, with INTEL i5-6600K processor and 8GB of RAM, running UBUNTU 18.04.

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Tropicalizing the Simplex Algorithm

Cited by 47 publications

References 40 publications

Lifting tropical bitangents

Lifting tropical bitangents

Tropical Analogues of a Dempe-Franke Bilevel Optimization Problem

Solving Mean-Payoff Games via Quasi Dominions

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