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47
Generating All Vertices of a Polyhedron Is Hard
 DISCRETE COMPUT GEOM (2008 ) 39 : 174–190 175
, 2008
"... We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other ne ..."
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Cited by 38 (6 self)
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We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P = NP. As a corollary, we solve in the negative two wellknown generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains
An Efficient Incremental Algorithm for Solving Systems of Linear Diophantine Equations
, 1994
"... In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several s ..."
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Cited by 35 (0 self)
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In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several subsystems. The proof of termination of the algorithm is difficult, whereas the proofs of completeness and correctness are straightforward generalizations of Fortenbacher's proof.
Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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Cited by 27 (1 self)
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
An exact duality theory for semidefinite programming based on sums of squares. ArXiv eprints
, 2012
"... Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebr ..."
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Cited by 12 (2 self)
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Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality A(x) ≽ 0 is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry. 1.
FARKASTYPE RESULTS WITH CONJUGATE FUNCTIONS
"... We present some new Farkastype results for inequality systems involving a finite as well as an infinite number of convex constraints. For this, we use two kinds of conjugate dual problems, namely an extended Fencheltype dual problem and the recently introduced FenchelLagrange dual problem. For t ..."
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Cited by 12 (4 self)
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We present some new Farkastype results for inequality systems involving a finite as well as an infinite number of convex constraints. For this, we use two kinds of conjugate dual problems, namely an extended Fencheltype dual problem and the recently introduced FenchelLagrange dual problem. For the latter, which is a ”combination” of the classical Fenchel and Lagrange duals, the strong duality is established.
Binary Linear Codes: New Results on Nonexistence
, 1996
"... ly, a constraint is a relation of the form a 1 v 1 + · · · +a n v n k, where v 1 , . . . , v n are variables, a 1 , . . . , a n # Q , k # Z, and is either #, #, or =. Constraints are represented in several different ways, according to what variables and coefficients are allowed: 1. Whe ..."
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Cited by 10 (0 self)
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ly, a constraint is a relation of the form a 1 v 1 + · · · +a n v n k, where v 1 , . . . , v n are variables, a 1 , . . . , a n # Q , k # Z, and is either #, #, or =. Constraints are represented in several different ways, according to what variables and coefficients are allowed: 1. When constraints appear in commands, the variables are arbitrary (within the confines of the language), and the coefficients are in Z. 2. These constraints are internally represented (using the "constraint" class) in the same way, except that Q coefficients are allowed. This is because some internal operations may result in constraints whose coefficients are not integers. (This can happen when the "incorporate" command is used.) 3. For split linear programming calculations, constraints are first converted to constraints involving v variables and having coefficients in Q (class qvconstraint). 4. Clearing denominators yields constraints having v variables and coefficients in Z (class vconstra...
Latticeordered rings and function rings
 Pacific J. Math
, 1962
"... Introduction: This paper treats the structure of those latticeordered rings which are subdirect sums of totally ordered rings—the frings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an importan ..."
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Cited by 9 (0 self)
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Introduction: This paper treats the structure of those latticeordered rings which are subdirect sums of totally ordered rings—the frings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an important overlap at the beginning.
ON THE DEVELOPMENT OF OPTIMIZATION THEORY
 THE AMERICAN MATHEMATICAL MONTHLY, 87 (1980), PP. 527{542.
, 1980
"... ..."
Potential Field Guide for Humanoid Multicontacts Acyclic Motion Planning
"... Abstract—We present a motion planning algorithm that computes rough trajectories used by a contactpoints planner as a guide to grow its search graph. We adapt collisionfree motion planning algorithms to plan a path within the guide space, a submanifold of the configuration space included in the fr ..."
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Cited by 7 (3 self)
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Abstract—We present a motion planning algorithm that computes rough trajectories used by a contactpoints planner as a guide to grow its search graph. We adapt collisionfree motion planning algorithms to plan a path within the guide space, a submanifold of the configuration space included in the free space in which the configurations are subject to static stability constraint. We first discuss the definition of the guide space. Then we detail the different techniques and ideas involved: relevant Cspace sampling for humanoid robot, taskdriven projection process, static stability test based on polyhedral convex cones theory’s double description method. We finally present results from our implementation of the algorithm. I.
Structured quadratic inverse eigenvalue problem, I. Serially linked systems
 SIAM J. Sci. Comput
"... Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constra ..."
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Cited by 6 (1 self)
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Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constraints must be respected. Consequently, they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation. Knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications. However, the issue of solvability is problem dependent and has to be addressed structure by structure. This paper considers one particular structure where the elements of the physical system, if modeled as a massspring system, are serially linked. The discussion recasts both undamped or damped problems in a framework of inequality systems that can be adapted for numerical computation. Some open questions are described. AMS subject classifications. 65F18, 15A22, 93B55