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61
The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
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Cited by 456 (0 self)
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The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floatingpoint arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.
polymake: a Framework for Analyzing Convex Polytopes
, 1999
"... polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few ..."
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Cited by 97 (15 self)
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polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tuberlin.de/diskregeom/polymake/.
Linear invariant generation using nonlinear constraint solving
 IN COMPUTER AIDED VERIFICATION
, 2003
"... We present a new method for the generation of linear invariants which reduces the problem to a nonlinear constraint solving problem. Our method, based on Farkas' Lemma, synthesizes linear invariants by extracting nonlinear constraints on the coefficients of a target invariant from a program. These ..."
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Cited by 64 (12 self)
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We present a new method for the generation of linear invariants which reduces the problem to a nonlinear constraint solving problem. Our method, based on Farkas' Lemma, synthesizes linear invariants by extracting nonlinear constraints on the coefficients of a target invariant from a program. These constraints guarantee that the linear invariant is inductive. We then apply existing techniques, including specialized quantifier elimination methods over the reals, to solve these nonlinear constraints. Our method has the advantage of being complete for inductive invariants. To our knowledge, this is the first sound and complete technique for generating inductive invariants of this form. We illustrate the practicality of our method on several examples, including cases in which traditional methods based on abstract interpretation with widening fail to generate sufficiently strong invariants.
The 01 Knapsack Problem With A Single Continuous Variable
 MATHEMATICAL PROGRAMMING
, 1997
"... Constraints arising in practice often contain many 01 variables and one or a small number of continuous variables. Existing knapsack separation routines cannot be used on such constraints. Here we study such constraint sets, and derive valid inequalities that can be used as cuts for such sets, as w ..."
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Cited by 34 (7 self)
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Constraints arising in practice often contain many 01 variables and one or a small number of continuous variables. Existing knapsack separation routines cannot be used on such constraints. Here we study such constraint sets, and derive valid inequalities that can be used as cuts for such sets, as well for more general mixed 01 constraints. Specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable, called the continuous 01 knapsack problem. First di#erent classes of facetdefining inequalities are derived based on projection and lifting. The order of lifting, particularly of the continuous variable, plays an important role. Secondly we show that the flow cover inequalities derived for the single node flow set, consisting of arc flows into and out of a single node with binary variable lower and upper bounds on each arc, can be obtained from valid inequalities for the continuous 01 knapsack problem. Thus the separation heuristic we derive for continuous knapsack sets can also be used to derive cuts for more general mixed 01 constraints. Initial computational results on a variety of problems are presented.
Constraintbased linearrelations analysis
 In Proc. SAS, LNCS 3148
, 2004
"... 1 Introduction Linearrelations analysis discovers linear relationships among the variables of aprogram that hold in all the reachable program states. Such relationships are called linear invariants. Invariants are useful in the verification of both safetyand liveness properties. Many existing techn ..."
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Cited by 30 (2 self)
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1 Introduction Linearrelations analysis discovers linear relationships among the variables of aprogram that hold in all the reachable program states. Such relationships are called linear invariants. Invariants are useful in the verification of both safetyand liveness properties. Many existing techniques rely on the presence of these invariants to prove properties of interest. Some types of analysis, e.g., variablebounds analysis, can be viewed as specializations of linearrelations analysis. Traditionally, this analysis is framed as an abstract interpretation in the domainof polyhedra [6, 7]. The analysis is carried out using a propagationbased technique, wherein increasingly accurate polyhedral iterates, converging towards thefinal result, are computed. This convergence is ensured through the use of widening, or extrapolation, operators. Such techniques are popular in the domains ofdiscrete and hybrid programs, motivating tools like
Exact volume computation for polytopes: a practical study
 in: Polytopes{combinatorics and computation (Oberwolfach
, 1997
"... ..."
Implementing generating set search methods for linearly constrained minimization
 Department of Computer Science, College of William and Mary
, 2005
"... Abstract. We discuss an implementation of a derivativefree generating set search method for linearly constrained minimization with no assumption of nondegeneracy placed on the constraints. The convergence guarantees for generating set search methods require that the set of search directions possess ..."
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Cited by 15 (4 self)
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Abstract. We discuss an implementation of a derivativefree generating set search method for linearly constrained minimization with no assumption of nondegeneracy placed on the constraints. The convergence guarantees for generating set search methods require that the set of search directions possesses certain geometrical properties that allow it to approximate the feasible region near the current iterate. In the hard case, the calculation of the search directions corresponds to finding the extreme rays of a cone with a degenerate vertex at the origin, a difficult problem. We discuss here how stateoftheart computational geometry methods make it tractable to solve this problem in connection with generating set search. We also discuss a number of other practical issues of implementation, such as the careful treatment of equality constraints and the desirability of augmenting the set of search directions beyond the theoretically minimal set. We illustrate the behavior of the implementation on several problems from the CUTEr test suite. We have found it to be successful on problems with several hundred variables and linear constraints.
Decomposition and Parallelization Techniques for Enumerating the Facets of 0/1Polytopes
 Int. J. Comput. Geom. Appl
, 1998
"... A convex polytope can either be described as convex hull of vertices or as solution set of a finite number of linear inequalities and equations. Whereas both representations are equivalent from a theoretical point of view, they are not when optimization problems over the polytope have to be solved. ..."
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Cited by 13 (2 self)
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A convex polytope can either be described as convex hull of vertices or as solution set of a finite number of linear inequalities and equations. Whereas both representations are equivalent from a theoretical point of view, they are not when optimization problems over the polytope have to be solved. Moreover, it is a challenging task in practical computation to convert one description into the other. In this paper we address the efficient computation of the facet structure of polytopes given by their vertices and present new computational results for polytopes which are of interest in combinatorial optimization. Keywords: polytope, convex hull, combinatorial optimization 1 Introduction Hard combinatorial optimization problems are often attacked with branchandcut methods. These methods strongly rely on knowledge about the structure of the polytope that is defined as convex hull of the 0/1 incidence vectors of feasible solutions. In particular, knowledge about linear equations and ineq...
Toward Autonomous FreeClimbing Robots
 IN INT. SYMP
, 2003
"... The goal of this research is to enable a multilimbed robot to climb vertical rock using techniques similar to those developed by human climbers. The robot consists of a small number of articulated limbs. Only the limb endpoints can make contact with the environmenta vertical surface with sma ..."
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Cited by 12 (10 self)
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The goal of this research is to enable a multilimbed robot to climb vertical rock using techniques similar to those developed by human climbers. The robot consists of a small number of articulated limbs. Only the limb endpoints can make contact with the environmenta vertical surface with small, arbitrarily distributed features called holds. A path through this environment is a sequence of onestep climbing moves in which the robot brings a limb endpoint to a new hold. The robot
Incremental Convex Hull Algorithms Are Not Output Sensitive
, 1996
"... A polytope is the bounded intersection of a finite set of halfspaces of R d . Every polytope can also be represented as the convex hull conv V of its vertices (or extreme points) V . The convex hull problem is to convert from the vertex representation to the halfspace representation or (equivalent ..."
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Cited by 10 (3 self)
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A polytope is the bounded intersection of a finite set of halfspaces of R d . Every polytope can also be represented as the convex hull conv V of its vertices (or extreme points) V . The convex hull problem is to convert from the vertex representation to the halfspace representation or (equivalently by geometric duality) viceversa. Given an ordering v 1 : : : vn of the input vertices, after some initialization an incremental convex hull algorithm constructs halfspace descriptions Hn\Gammak : : : Hn where H i is the halfspace description of convf v 1 : : : v i g. Let m i denote jH i j, and let m denote mn . Let OE(d) denote d=d p d e \Gamma 1; in this paper we give families of polytopes for which mn\Gamma1 2 \Omega\Gamma m OE(d) ) for any ordering of the input. We also give a family of 0=1polytopes with a similar blowup in intermediate size. Since mn\Gamma1 is not bounded by any polynomial in m, n, and d, incremental convex hull algorithms cannot in any reasonable sense be con...