Book•
Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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Dissertation•
06 Jul 2015
TL;DR: Deza et al. as discussed by the authors showed that computing a Nash equilibrium in a bimatrix game is polynomially reducible to a colorful linear programming problem, which is a variant of the "Barany-Onn" algorithm.
Abstract: The colorful Caratheodory theorem, proved by Barany in 1982, states the following. Given d A1 sets of points S1, . . . ,SdA1 µ Rd , each of them containing 0 in its convex hull, there exists a colorful set T containing 0 in its convex hull, i.e. a set T µ SdA1 iAE1 Si such that jT \Si j • 1 for all i and such that 0 2 conv(T ). This result gave birth to several questions, some algorithmic and some more combinatorial. This thesis provides answers on both aspects. The algorithmic questions raised by the colorful Caratheodory theorem concern, among other things, the complexity of finding a colorful set under the condition of the theorem, and more generally of deciding whether there exists such a colorful set when the condition is not satisfied. In 1997, Barany and Onn defined colorful linear programming as algorithmic questions related to the colorful Caratheodory theorem. The two questions we just mentioned come under colorful linear programming. This thesis aims at determining which are the polynomial cases of colorful linear programming and which are the harder ones. New complexity results are obtained, refining the sets of undetermined cases. In particular, we discuss some combinatorial versions of the colorful Caratheodory theorem from an algorithmic point of view. Furthermore, we show that computing a Nash equilibrium in a bimatrix game is polynomially reducible to a colorful linear programming problem. On our track, we found a new way to prove that a complementarity problem belongs to the PPAD class with the help of Sperner’s lemma. Finally, we present a variant of the “Barany-Onn” algorithm, which is an algorithmcomputing a colorful set T containing 0 in its convex hull whose existence is ensured by the colorful Caratheodory theorem. Our algorithm makes a clear connection with the simplex algorithm. After a slight modification, it also coincides with the Lemke method, which computes a Nash equilibriumin a bimatrix game. The combinatorial question raised by the colorful Caratheodory theorem concerns the number of positively dependent colorful sets. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597–604 (2006)) conjectured that, when jSi j AE d A1 for all i 2 {1, . . . ,d A1}, there are always at least d2A1 colourful sets containing 0 in their convex hulls. We prove this conjecture with the help of combinatorial objects, known as the octahedral systems. Moreover, we provide a thorough study of these objects
1 citations
Cites background from "Linear complementarity, linear and ..."
...There are classes of matrices M for which the algorithm reaches an infinite ray if and only if the problem LCP(M ,q) is infeasible, see [44] for more details on this subject....
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Dissertation•
01 Jan 2014
1 citations
Cites methods from "Linear complementarity, linear and ..."
...If the problem at hand contains only linear constraints then it can be solved as presented by Murty [65] in the linear case and as presented by Levendovszky et al [54] in the discrete binary case....
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...Journal of Chemical Physics, 21(6), 1087-1092 [65] Murty, K....
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Dissertation•
01 May 2012
1 citations
Cites methods from "Linear complementarity, linear and ..."
...9 can be transformed to a linear complementarity problem (LCP) (q,M) [70] whose feasible solution can be obtained using algorithms found in literature like PATH [71]....
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Dissertation•
01 Dec 2017
TL;DR: In this paper, the authors propose to use semelles exterieures souples (c'est-a-dire deformables) aux pieds to absorb les impacts and limitent les effets des irregularites du sol pendant le mouvement.
Abstract: Lorsque des changements inattendus de la surface du sol se produisent lors de la marche, le systeme nerveux central humain doit appliquer des mesures de controle appropriees pour assurer une stabilite dynamique. De nombreuses etudes dans le domaine de la commande moteur ont etudie les mecanismes d'un tel controle postural et ont largement decrit comment les trajectoires du centre de masse (COM), le placement des pas et l'activite musculaire s'adaptent pour eviter une perte d'equilibre. Les mesures que nous avons effectuees montrent qu'en arrivant sur un sol mou, les participants ont module de facon active les forces de reaction au sol (GRF) sous le pied de support afin d'exploiter les proprietes elastiques et deformables de la surface pour amortir l'impact et probablement dissiper l'energie mecanique accumulee pendant la ‘chute’ sur la nouvelle surface deformable. Afin de controler plus efficacement l'interaction pieds-sol des robots humanoides pendant la marche, nous proposons d'ajouter des semelles exterieures souples (c'est-a-dire deformables) aux pieds. Elles absorbent les impacts et limitent les effets des irregularites du sol pendant le mouvement sur des terrains accidentes. Cependant, ils introduisent des degres de liberte passifs (deformations sous les pieds) qui complexifient les tâches d'estimation de l'etat du robot et ainsi que sa stabilisation globale. Pour resoudre ce probleme, nous avons concu un nouveau generateur de modele de marche (WPG) base sur une minimisation de la consommation d'energie qui genere les parametres necessaires pour utiliser conjointement un estimateur de deformation base sur un modele elements finis (FEM) de la semelle souple pour prendre en compte sa deformation lors du mouvement. Un tel modele FEM est couteux en temps de calcul et empeche la reactivite en ligne. Par consequent, nous avons developpe une boucle de controle qui stabilise les robots humanoides lors de la marche avec des semelles souples sur terrain plat et irregulier. Notre controleur en boucle fermee minimise les erreurs sur le centre de masse (COM) et le point de moment nul (ZMP) avec un controle en admittance des pieds base sur un estimateur de deformation simplifie. Nous demontrons son efficacite experimentalement en faisant marcher le robot humanoide HRP-4 sur des graviers.
1 citations