T. S. Motzkin, H. Raiffa et al. The double description method. Theodore S. Motzkin: Selected Papers, 1953.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 Introduction Many applications of static analysis and verification compute on some abstract domain based on convex polyhedra [5] Traditionally, most of these applications are restricted to convex polyhedra that are topologically closed. When adopting the Double Description (DD) method [10], a closed convex polyhedron can be specified in two ways, using a constraint system or a generator system: the constraint system contains a finite set of linear non strict inequality constraints; the generator system contains two finite sets of vectors, collectively called generators, which are ....

T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The double description method. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games -- Volume II, number 28 in Annals of Mathematics Studies, pages 51--73. Princeton University Press, Princeton, New Jersey, 1953.

....vertices of the polyhedron [3] for instance, any half space of R has two extreme rays and no vertices, but any generator system describing it will contain at least three rays and one point. The combination of the two approaches outlined above is the basis of the Double Description (DD) method [30], which exploits the duality principle to compute each representation starting from the other one, possibly minimizing the descriptions. We will write con(C) and gen(G) to denote the polyhedra described by the finite constraint system respectively. Definition 2. DD pair and minimal forms. ....

T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The double description method. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games -- Volume II, number 28 in Annals of Mathematics Studies, pages 51--73. Princeton University Press, Princeton, New Jersey, 1953. 20

.... (1) has a solution of complexity O(k bn=2c ) where k is the number of inequality constraints, n the dimension of the space, and the input length is Omega Gamma kn) 1] This algorithm was implemented in the Polyhedral Library by Wilde [16] It is based on the Motzkin double description method [12], and builds on the work done by Fern andez and Quinton [5] and Le Verge [8] 2.2 Faces of a polyhedron Definition 3. A supporting hyperplane of an n polyhedron D is a plane of dimension n Gamma 1 which intersects the hull of D but not its relative interior. Definition 4. A face of a ....

Motzkin, T. S., Raiffa, H., L. Thompson, G. and Thrall, R. M. The double description method., 1953.

....1 Introduction Many applications of static analysis and verification compute on some abstract domain based on convex polyhedra. Traditionally, most of these applications are restricted to convex polyhedra that are topologically closed. When adopting the Double Description (DD) method [8], a closed convex polyhedron can be specified in two ways, using a constraint system or a generator system: the constraint system contains a finite set of linear non strict inequality constraints; the generator system contains two finite sets of vectors, collectively called generators, which are ....

T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The double description method. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games -- Volume II, number 28 in Annals of Mathematics Studies, pages 51--73. Princeton University Press, Princeton, New Jersey, 1953.

....1 Introduction Many applications of static analysis and verification compute on some abstract domain based on convex polyhedra. Traditionally, most of these applications are restricted to convex polyhedra that are topologically closed. When adopting the Double Description (DD) method [8], a closed convex polyhedron can be specified in two ways, using a constraint system or a generator system: the constraint system contains a finite set of linear non strict inequality constraints; the generator system contains two finite sets of vectors, collectively called generators, which are ....

T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The double description method. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games -- Volume II, number 28 in Annals of Mathematics Studies, pages 51--73. Princeton University Press, Princeton, New Jersey, 1953.

....vertices of the polyhedron [3] for instance, any half space of R has two extreme rays and no vertices, but any generator system describing it will contain at least three rays and one point. The combination of the two approaches outlined above is the basis of the Double Description (DD) method [29], which exploits the duality principle to compute each representation starting from the other one, possibly minimizing the descriptions. We will write con(C) and gen(G) to denote the polyhedra described by the finite constraint system G, respectively. Definition 2. DD pair and minimal ....

T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The double description method. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games -- Volume II, number 28 in Annals of Mathematics Studies, pages 51--73. Princeton University Press, Princeton, New Jersey, 1953.

....each one given by a linear inequality (H representation) This equivalence is known as the Weyl Minkowski theorem (e.g. 32, p. 29] The problem to obtain all inequalities from the vertices of a convex polytope is known as the hull problem. One solution strategy is the Double Description Method [33] which we shall use but not review here. What Boole did not foresee, however, is that certain events in one and the same inequality may be operationally incompatible, and that the event structure may not be a Boolean algebra. This applies to quantum mechanics as well as to Wright s generalized ....

T.S. Motzkin, H. Raia, G.L. Thompson, and R.M. Thrall. The double description method. In Contributions to theory of games, Vol. 2. Princeton University Press, New Jersey, Princeton, 1953.

....of half spaces, each one given by a linear inequality (H representation) This equivalence is known as the Weyl Minkowski theorem. The problem to obtain all inequalities from the vertices of a convex polytope is known as the hull problem. One solution strategy is the Double Description Method [12] which we shall use but not review here. 1.4 From vertices to inequalities For the above simple urn model, the inequalities are rather intuitive and can be easily obtained by guessing. This is impossible in the general case involving more events and more joint probabilities thereof. In order to ....

....the frequent use of this function one can append this command to the package file cddif.m before the line End[ so that on each loading of the package the directory is set automatically to a personal working directory. 2. 2 cdd cdd is a C (ANSI C) implementation of the Double Description Method [12] by Komei Fukuda[14] It generates all vertices (i.e. extreme points) and extreme rays of a general convex polyhedron given by a system of linear inequalities. Conversely, it solves the hull problem by generating a system of linear inequalities given all vertices. At this point we refer to the ....

T.S. Motzkin, H. Raiffa, G.L. Thompson, and R.M. Thrall. The double description method. In Contributions to theory of games, Vol. 2. Princeton University Press, New Jersey, Princeton, 1953.

....application of these two rules will guarantee that the resulting system is free of redundancy. 4.3. The double description method As Matheis and Rubin[1980] point out, all nonpivoting methods for transforming the description of polyhedra can be viewed as variants of the doubledescription method (Motzkin, Thomson, Raiffa and Thrall[1953]) which transforms H to V representations of polyhedral cones. This method can best be explained by geometric visualization. Suppose we have a cone P with known minimum inner description. Let P 0 be obtained from P by adding one further constraint. This constraint de nes a hyperplane H and a ....

Motzkin, T.S, H. Raia, G.L. Thomson and R.M. Thrall (1953). The Double Description Method, in H.W. Kuhn and A.W. Tucker, eds. Contributions to the Theory of Games (Vol. 2). Princeton University Press, Princeton, N.J..

....for Enumerating All Schedules Given this background, the approach we use to enumerate all schedules can now be spelled out 4 . The algorithm has 3 basic parts as given in Figures 3.7 to 3.9 in a mixture of C and English. Roughly 4 The algorithm is similar to one given by Foutrier and Motzkin [29] 36 0 af 1 bge 2 ch f 3 d ga 4 i hbe 5 c af 6 j d bg 7 i ceh 8 j d af 9 i bge 10 ch f 11 j d ga 12 i hbe 13 c 14 j d 15 i 16 j Figure 3.6: Loop kernel for alternate schedule for example loop 37 speaking, these are ....

Theodore S. Motzkin. The Double Description Method. In G.L. Thompson T.S. Motzkin, H. Raiffa and R.M. Thrall, editors, Theodore S. Motzkin: Selected Papers. 1953.

....many commercial codes and public codes available. See the LP FAQ [FG] Two excellent books on LP are Chvatal s textbook [Chv83] and Schrijver s researcher s bible [Sch86] 5 Polyhedral Computation Codes cddlib, cdd and cdd [Fuk95] C and C implementations of the double description method [MRTT53]) Comments: Floating and exact arithmetic. Ecient for highly degenerate cases. The exact version cddr is much slower. It can remove redundancies from input data using a built in LP code. cddlib is a C library with basic polyhedral conversion functions and LP solvers. cddlib can be compiled ....

T.S. Motzkin, H. Raia, GL. Thompson, and R.M. Thrall. The double description method. In H.W. Kuhn and A.W.Tucker, editors, Contributions to theory of games, Vol. 2. Princeton University Press, Princeton, RI, 1953.

....if e 2 E then Pe 6= 0. We have that u is a solution of (4) 5) if and only if it can be written as u = X c i 2C i c i X e i 2E i e i ; 6) with i 2 R and i 0. To calculate the sets C and E we use an iterative algorithm that is an adaptation of the double description method of Motzkin (Motzkin, Rai a, Thompson and Thrall, 1953). During the iteration we already remove rays that do not satisfy the partial complementarity condition since such rays cannot yield solutions of the ELCP. In the kth step the partial complementarity condition is de ned as follows: Y i2 j (Pu) i = 0; 8j 2 f1; 2; mg such that j f1; ....

Motzkin, T.S., Raia, H., Thompson, G.L., and Thrall, R.M. 1953. The double description method. In Contributions to the theory of games, no. 28 in Annals of Math. Studies. Princeton: Princeton University Press, pp. 51-73.

.... efficient tools were necessary to provide computational kernel on convex polyhedra ( Pug92, Fea92, Bal95, Tei93, Wil93] Manipulating upon these structures is not easy, and the mathematical foundation of the so called polyhedral theory, has its roots in theory of linear and integer programming ([MRTT53, Che68, Sch86, NW88]) A convex polyhedron has two dual representations: intersection of half spaces or convex combinations of vertices. These representations were introduced by Motzkin ( MRTT53] and Chernikova showed how to go from one to the other ( Che68] Successive improvements of the Chernikova algorithm ....

....of the so called polyhedral theory, has its roots in theory of linear and integer programming ( MRTT53, Che68, Sch86, NW88] A convex polyhedron has two dual representations: intersection of half spaces or convex combinations of vertices. These representations were introduced by Motzkin ([MRTT53]) and Chernikova showed how to go from one to the other ( Che68] Successive improvements of the Chernikova algorithm ( FQ88, Ver92] provided a very efficient computational kernel on convex polyhedra. Using this kernel, Wilde implemented a public domain polyhedral library ( Wil93] which ....

T.S. Motzkin, H. Raiffa, G.L. Thompson, and R.M. Thrall. The double description method. Theodore S. Motzkin: Selected Papers, 1953.

.... In higher dimensions, 4 Delta the best output sensitive algorithm is Seidel s shelling algorithm at O(n 2 h log n) when h = Omega Gamma n) Seidel 1986] and gift wrapping at O(nh) otherwise [Chand and Kapur 1970] The Double Description Method is the dual of the Beneath Beyond Algorithm [Motzkin et al. 1953]. It is the earliest incremental method for computing the convex hull. It is an excellent choice in high dimensions when the number of facets is much smaller than the maximum number of facets for r vertices (f r ) Avis and Bremner 1995] Fukuda and Prodon 1996] 2. THE QUICKHULL ALGORITHM We ....

Motzkin, T. S., Raiffa, H., Thompson, G. L., and Thrall, R. M. 1953. The double description method. In H. W. Kuhn and A. W. Tucker Eds., Contributions to the Theory of Games II , Volume 8 of Annals of Mathematical Studies, pp. 51--73. Princeton University Press.

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T. S. Motzkin, H. Raiffa et al. The double description method. Theodore S. Motzkin: Selected Papers, 1953.

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T.S. Motzkin, H. Raiffa, G.L. Thompson, and R.M. Thrall. The double description method. Theodore S. Motzkin: Selected Papers, 1953. reprinted in D.Cantor, B. Gordon and B. Rothschild, eds, Birkhauser, Boston, 1983.

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T. S. Motzkin, H. Rai#a, G. L. Thompson, R. M. Thrall, The double description method, in: H. W. Kuhn, A. W. Tucker (Eds.), Contributions to the Theory of Games -- Volume II, no. 28 in Annals of Mathematics Studies, Princeton University Press, Princeton, New Jersey, 1953, pp. 51--73.

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T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The double description method. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games -- Volume II, number 28 in Annals of Mathematics Studies, pages 51--73. Princeton University Press, Princeton, New Jersey, 1953.

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T.S. Motzkin, H. Raiffa, G.L. Thompson and R.M. Thrall. The double description method. Theodore S. Motzkin: Selected Papers, 1953.

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T. S. Motzkin, H. Rai#a, G. L. Thompson, R. M. Thrall, The double description method, in: H. W. Kuhn, A. W. Tucker (Eds.), Contributions to the Theory of Games -- Volume II, no. 28 in Annals of Mathematics Studies, Princeton University Press, Princeton, New Jersey, 1953, pp. 51--73.

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T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The Double Description Method. In Contributions to the Theory of Games, number 28 in Annals of Mathematics Study. Princeton University Press, 1953.

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T. S. Motzkin, H. Rai#a, G. L. Thompson, and R. M. Thrall. The double description method. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games -- Volume II, number 28 in Annals of Mathematics Studies, pages 51--73. Princeton University Press, Princeton, New Jersey, 1953.

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T.S. Motzkin, H. Raiffa, G.L. Thompson, and R.M. Thrall. The double description method. In Contributions to theory of games, Vol. 2. Princeton University Press, New Jersey, Princeton, 1953.

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T.S. Motzkin, H. Raiffa, G.L. Thompson, R.M. Thrall, The double description method, Ann. of Math. Stud. 8 (1953) 51-- 73.

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Motzkin, T.S.; Raiffa, H.; Thompson, G.L.; and Thrall, R.M., "The Double Description Method", in Contributions to the Theory of Games Vol. 2, H.W. Kuhn and A.W.Tucker, (Eds.), Princeton University Press, Princeton, NJ, 1953.

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