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Small-dimensional linear programming and convex hulls made easy

1991
*
Discrete & Computational Geometry
*

One solves

doi:10.1007/bf02574699
fatcat:r7enleuw3reohlgn7y3f5kvhza
*linear**programs*involving m constraints in d variables in expected time O(m). The other constructs*convex**hulls*of n points in Nd, d > 3, in expected time O(nln/2l). ... In the*linear**programming*algorithm the dependence of the time bound on d is of the form d!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses. ...*Convex**Hulls*This section concerns the construction of the*convex**hull*of a set S of n points in Ne. ...##
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A fast linear separability test by projection of positive points on subspaces

2007
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Proceedings of the 24th international conference on Machine learning - ICML '07
*

The worst case computational complexity of our algorithm is lower than the worst case computational complexity of Simplex, Perceptron, Support Vector Machine

doi:10.1145/1273496.1273586
dblp:conf/icml/YoganandaMG07
fatcat:y6rl3vw7wbgyfb47sbmvojel6m
*and**Convex**Hull*Algorithms, if d < n 2 3 . ... The worst case time complexity of the algorithm is O(nr 3 )*and*space complexity of the algorithm is O(nd). A*small*review of some of the prominent algorithms*and*their time complexities is included. ... Algorithms A general discussion of*Linear**Programming*, Quadratic*Programming*,*Convex**Hull**and*the Perceptron Algorithms to solve the*linear*separability problem is included in this subsection. ...##
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Data-based polyhedron model for optimization of engineering structures involving uncertainties

2021
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Data-Centric Engineering
*

Then the vertex solution of

doi:10.1017/dce.2021.8
fatcat:t5ucgct6rrgetb6gaw6qrphx3q
*convex*polyhedron*linear**programming*is presented*and*proven. ... Secondly, the characteristics of the optimal solution of*convex*polyhedron*linear**programming*are investigated. ... Similar to example 1, it is*easy*to prove that G is a*convex*polyhedron. Therefore (57) is a standard form of a*convex*polyhedron*linear**programming*problem. ...##
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Selecting texture discriminative descriptors of capsule endpscopy images

2009
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2009 Proceedings of 6th International Symposium on Image and Signal Processing and Analysis
*

In this paper we propose a new method for feature subset selection utilizing a

doi:10.1109/ispa.2009.5297634
fatcat:ixcp7n47ujf7tavu2rv5mjnx3y
*convex**hull*(or*convex*polytope). ... In supervised data classification one of the problems is to reduce*dimensionality*of feature vectors. ... The next step is to find a centroid c of the*convex**hull*. Then the*convex**hull*is isotropically scaled up (cf. Fig. 1c) around the fixed centroid c of the*convex**hull*. ...##
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Learning to rank using 1-norm regularization and convex hull reduction

2010
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Proceedings of the 48th Annual Southeast Regional Conference on - ACM SE '10
*

However this approach increases the problem complexity from

doi:10.1145/1900008.1900052
dblp:conf/ACMse/NanCDW10
fatcat:tzkeynse5jg2bkebyc7osgycdq
*linear*to quadratic in terms of sample size. We present in this paper a*convex**hull*reduction method to reduce this impact. ... We also propose a 1-norm regularization approach to simultaneously find a*linear*ranking function*and*to perform feature subset selection. The proposed method is formulated as a*linear**program*. ... Any opinions, findings*and*conclusions or recommendations expressed in this material are those of the authors*and*do not necessarily reflect those of the National Science Foundation. ...##
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Human-machine interaction for real-time linear optimization

2012
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2012 IEEE International Conference on Systems, Man, and Cybernetics (SMC)
*

Therefore, we propose a system providing the decision-makers with a

doi:10.1109/icsmc.2012.6377804
dblp:conf/smc/HamelGQBM12
fatcat:arzkzixr6zcpbf4cnkytgiluwi
*convex**hull*of optimal solutions minimizing/maximizing the variables of interest. ... These problems can be solved in less than an hour as they show a*linear*structure. ... Its*linear*optimization problem is*small**and**easy*to solve -although it is of not as efficient as the heuristic methods. However, those three approaches meat our real-time expectations. ...##
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A New Algorithm for 3D Reconstruction from Support Functions

2009
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IEEE Transactions on Pattern Analysis and Machine Intelligence
*

In addition we offer a

doi:10.1109/tpami.2008.190
pmid:19147881
fatcat:6kj3suop6ng2rf4zm2qjbidcgq
*linear**program*version of the new algorithm that is much faster*and*better, or at least comparable, in performance at low levels of noise*and*reasonably*small*numbers of measurements ... The algorithm, based on a least squares procedure, is very*easy*to*program*in standard software such as Matlab,*and*it works for both 2D*and*3D reconstructions (in fact, in principle, in any dimension) ... Then it is*easy*to check thatx 1 , . . . ,x k is also a solution of (8)- (9) ,*and*the*convex**hull*of {x 1 , . . . ,x k } is a possible output of the new support function algorithm. ...##
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Support vector machines

2000
*
SIGKDD Explorations
*

The classification problem is used to investigate the basic concepts behind SVMs

doi:10.1145/380995.380999
fatcat:mwuco6xjejhrjmcj7o4clqy6da
*and*to examine their strengths*and*weaknesses from a data mining perspective. ... A B S T R A C T Support Vector Machines (SVMs)*and*related kernel methods have become increasingly popular tools for data mining tasks such as classification, regression,*and*novelty detection. ... This work was performed with the support of the National Science Foundation under grants 970923*and*IIS-9979860. ...##
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Computing convex hulls and counting integer points with polymake

2016
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Mathematical Programming Computation
*

The main purpose of this paper is to report on the state of the art of computing integer

doi:10.1007/s12532-016-0104-z
fatcat:qk3iih53zfc3fdscu54lh6wwey
*hulls**and*their facets as well as counting lattice points in*convex*polytopes. ... Using the polymake system we explore various algorithms*and*implementations. Our experience in this area is summarized in ten "rules of thumb". ... The comments by David Avis*and*Winfried Bruns were particularly detailed. ...##
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Geometric Reasoning with polymake
[article]

2005
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arXiv
*
pre-print

The mathematical software system polymake provides a wide range of functions for

arXiv:math/0507273v1
fatcat:tvougotxu5a73maybpwesr4p3q
*convex*polytopes, simplicial complexes,*and*other objects. ... Later sections include a survey of research results obtained with the help of polymake so far*and*a short description of the technical background. ...*Linear**programming*. Polytopes most naturally appear as sets of feasible solutions of*linear**programs*. Consider the following example. Figure 2 . 2*Small**linear**program**and*a visualization. ...##
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Sequential convexification in reverse convex and disjunctive programming

1989
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Mathematical programming
*

each time the

doi:10.1007/bf01587096
fatcat:ezvkwghrt5dl7gmi4wzfpijeiu
*convex**hull*of the resulting set. ... Here weextend the class of problems considered to disjunctive*programs*with infinitely many terms, also known as reverse*convex**programs*,*and*give necessary*and*sufficient conditions for the ,-." solution ... In a disjunctive 'K*programming*problem with finitely many terms in each disjunction, each g. is either*linear*or piecewise*linear**and**convex*. ...##
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Star splaying

2005
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Proceedings of the twenty-first annual symposium on Computational geometry - SCG '05
*

Star splaying is a general-

doi:10.1145/1064092.1064129
dblp:conf/compgeom/Shewchuk05
fatcat:yopblutnv5acjfgvun26cwvnq4
*dimensional*algorithm that takes as input a triangulation or an approximation of a*convex**hull*,*and*produces the Delaunay triangulation, weighted Delaunay triangulation, or*convex*...*linear*in the number of vertices. ... Acknowledgments My thanks go to François Labelle for debunking a previous failed attempt at these results,*and*to Raimund Seidel for helpful discussions*and*for sharing his O( √ n) point location method ...##
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Likelihood-based Inference for Exponential-Family Random Graph Models via Linear Programming
[article]

2022
*
arXiv
*
pre-print

While the

arXiv:2202.03572v1
fatcat:hjzvdg5icjdurf45piy76gitvq
*convex**hull*question may be solved via a simple*linear**program*, this approach is not well known in the statistical literature. ... This article discusses the problem of determining whether a given point, or set of points, lies within the*convex**hull*of another set of points in d dimensions. ... To reiterate, the*convex**hull*of any set of d-*dimensional*points is the smallest*convex*set containing that set,*and*the*convex**hull*is always closed in R d . ...##
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On Functional Separately Convex Hulls

1998
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Discrete & Computational Geometry
*

Zhang on the existence of higher-

doi:10.1007/pl00009331
fatcat:gwcdua7ce5b5pdlqt6sx7fxoy4
*dimensional*nontrivial configurations of points*and*matrices). ... We prove some results concerning the structure of functional D-*convex**hulls*, e.g., a Krein-Milman-type theorem*and*a result on separation of connected components. ... We also thank Pankaj Agarwal, Derick Wood,*and*Nati Linial for pointing out*and*/or providing relevant literature. ...##
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Convex Hulls in a 3-Dimensional Space
[chapter]

2004
*
Lecture Notes in Computer Science
*

This paper describes a new algorithm of computing the

doi:10.1007/978-3-540-30503-3_14
fatcat:22uhww5renh6tk3lnsx3ef2aaq
*convex**hull*of a 3-*dimensional*object. ... The*convex**hull*generated by this algorithm is an abstract polyhedron being described by a new data structure, the cell list, suggested by one of the authors. ... After sorting, the*convex**hull*is computed by a recursive function consisting of two parts: generation of the*convex**hull*of a*small*subset of points*and*merging two*convex**hulls*. ...
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