Results 1  10
of
14
On the Complexity of Computing Estimates of Condition Measures of a Conic Linear System
, 2001
"... Condition numbers based on the "distance to illposedness" (d) have been shown to play a crucial role in the theoretical complexity of solving convex optimization models. In this paper we present two algorithms and corresponding complexity analysis for computing estimates of (d) for a finitedimensi ..."
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Cited by 14 (9 self)
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Condition numbers based on the "distance to illposedness" (d) have been shown to play a crucial role in the theoretical complexity of solving convex optimization models. In this paper we present two algorithms and corresponding complexity analysis for computing estimates of (d) for a finitedimensional convex feasibility problem P (d) in standard primal form: find x that satisfies Ax = b; x 2 CX , where d = (A; b) is the data for the problem P (d). Under one choice of norms for the m and n dimensional spaces, the problem of estimating (d) is hard (coNP complete even when CX = < n + ). However, when the norms are suitably chosen, the problem becomes much easier: we can estimate (d) to within a constant factor of its true value with complexity bounds that are linear in ln(C(d)) (where C(d) is the condition number of the data d for P (d)), plus other quantities that arise naturally in consideration of the problem P (d). The first algorithm is an interiorpoint algorithm, and the second algorithm is a variant of the ellipsoid algorithm. The main conclusion of this work is that when the norms are suitably
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 12 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
Set Containment Characterization
 ICM 2006 – Madrid
, 2006
"... Characterization of the containment of a polyhedral set in a closed halfspace, a key factor in generating knowledgebased support vector machine classifiers [7], is extended to the following: (i) Containment of one polyhedral set in another. (ii) Containment of a polyhedral set in a reverseconvex ..."
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Cited by 5 (0 self)
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Characterization of the containment of a polyhedral set in a closed halfspace, a key factor in generating knowledgebased support vector machine classifiers [7], is extended to the following: (i) Containment of one polyhedral set in another. (ii) Containment of a polyhedral set in a reverseconvex set defined by convex quadratic constraints. (iii) Containment of a general closed convex set, defined by convex constraints, in a reverseconvex set defined by convex nonlinear constraints. The first two characterizations can be determined in polynomial time by solving m linear programs for (i) and m convex quadratic programs for (ii), where m is the number of constraints defining the containing set. In (iii), m convex programs need to be solved in order to verify the characterization, where again m is the number of constraints defining the containing set. All polyhedral sets, like the knowledge sets of support vector machine classifiers, are characterized by the intersection of a finite number of closed halfspaces. Keywords set containment, knowledgebased classifier, linear programming, quadratic programming 1
Novel Approaches To The Discrimination Problem
, 1992
"... this paper we follow an altogether different approach (see also Cavalier et al. [3]). We do not assume any functional form for the distributions FX and F Y . Actually we do not even assume the points x ..."
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Cited by 4 (0 self)
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this paper we follow an altogether different approach (see also Cavalier et al. [3]). We do not assume any functional form for the distributions FX and F Y . Actually we do not even assume the points x
Convex Hulls, Oracles, and Homology
 J. SYMBOLIC COMPUT
, 2004
"... This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plu ..."
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Cited by 4 (0 self)
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This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the “no”case of CompletenessC has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).
On the hardness of computing intersection, union and minkowski sum of polytopes
 DISCRETE & COMPUTATIONAL GEOMETRY
"... For polytopes P1, P2 ⊂ R d we consider the intersection P1 ∩ P2, the convex hull of the union CH(P1 ∪ P2), and the Minkowski sum P1 + P2. For Minkowski sum we prove that enumerating the facets of P1+P2 is NPhard if P1 and P2 are specified by facets, or if P1 is specified by vertices and P2 is a poly ..."
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Cited by 3 (1 self)
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For polytopes P1, P2 ⊂ R d we consider the intersection P1 ∩ P2, the convex hull of the union CH(P1 ∪ P2), and the Minkowski sum P1 + P2. For Minkowski sum we prove that enumerating the facets of P1+P2 is NPhard if P1 and P2 are specified by facets, or if P1 is specified by vertices and P2 is a polyhedral cone specified by facets. For intersection we prove that computing the facets or the vertices of the intersection of two polytopes is NPhard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NPhard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NPcomplete.
Polyhedral Boundary Projection
, 1998
"... . We consider the problem of projecting a point in a polyhedral set onto the boundary of the set using an arbitrary norm for the projection. Two types of polyhedral sets, one defined by a convex combination of k points in R n and the second by the intersection of m closed halfspaces in R n , lea ..."
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Cited by 2 (0 self)
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. We consider the problem of projecting a point in a polyhedral set onto the boundary of the set using an arbitrary norm for the projection. Two types of polyhedral sets, one defined by a convex combination of k points in R n and the second by the intersection of m closed halfspaces in R n , lead to disparate optimization problems for finding such a projection. The first case leads to a mathematical program with a linear objective function and constraints that are linear inequalities except for a single nonconvex cylindrical constraint. Interestingly, for the 1norm, this nonconvex problem can be solved by solving 2n linear programs. The second polyhedral set leads to a much simpler problem of determining the minimum of m easily evaluated numbers. These disparate mathematical complexities parallel known ones for the related problem of finding the largest ball, with radius measured by an arbitrary norm, that can be inscribed in the polyhedral set. For a polyhedral set of the first t...
On a cone covering problem
 In Proceedings of the 20th Annual Canadian Conference on Computational Geometry
, 2008
"... Given a set of polyhedral cones C1, · · · , Ck ⊂ Rd, and a convex set D, does the union of these cones cover the set D? In this paper we consider the computational complexity of this problem for various cases such as whether the cones are defined by extreme rays or facets, and whether D is the en ..."
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Cited by 2 (0 self)
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Given a set of polyhedral cones C1, · · · , Ck ⊂ Rd, and a convex set D, does the union of these cones cover the set D? In this paper we consider the computational complexity of this problem for various cases such as whether the cones are defined by extreme rays or facets, and whether D is the entire Rd or a given linear subspace Rt. As a consequence, we show that it is coNPcomplete to decide if the union of a given set of convex polytopes is convex, thus answering a question of Bemporad, Fukuda and Torrisi. 1.
On Computing the Shadows and Slices of polytopes
, 2008
"... We study the projection of polytopes along k orthogonal vectors for various input and output forms. We show that if k is part of the input and we are interested in outputsensitive algorithms, then in most forms the problem is equivalent to enumerating vertices of polytopes, except in two where it i ..."
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Cited by 1 (1 self)
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We study the projection of polytopes along k orthogonal vectors for various input and output forms. We show that if k is part of the input and we are interested in outputsensitive algorithms, then in most forms the problem is equivalent to enumerating vertices of polytopes, except in two where it is NPhard. In two other forms the problem is trivial. We also review the complexity of computing projections when the projection directions are picked at random. For fulldimensional polytopes containing origin in the interior, projection is an operation dual to intersecting the polytope with a suitable hyperplane and so the results in this paper can be dualized by interchanging vertices with facets and projection with intersection. We would like to remark that even though most of the results in this paper do not appear to have been published before, they follow from straighforward reductions to other known results. The purpose of this paper is to serve as a reference to these results about the computational complexity of projection of polytopes onto affine subspaces.