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Small-dimensional linear programming and convex hulls made easy

Raimund Seidel
1991 Discrete & Computational Geometry  
One solves 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.  ... 
doi:10.1007/bf02574699 fatcat:r7enleuw3reohlgn7y3f5kvhza

A fast linear separability test by projection of positive points on subspaces

Yogananda A P, M Narasimha Murthy, Lakshmi Gopal
2007 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 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.  ... 
doi:10.1145/1273496.1273586 dblp:conf/icml/YoganandaMG07 fatcat:y6rl3vw7wbgyfb47sbmvojel6m

Data-based polyhedron model for optimization of engineering structures involving uncertainties

Zhiping Qiu, Han Wu, Isaac Elishakoff, Dongliang Liu
2021 Data-Centric Engineering  
Then the vertex solution of 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.  ... 
doi:10.1017/dce.2021.8 fatcat:t5ucgct6rrgetb6gaw6qrphx3q

Selecting texture discriminative descriptors of capsule endpscopy images

P. Szczypinski, A. Klepaczko
2009 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 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.  ... 
doi:10.1109/ispa.2009.5297634 fatcat:ixcp7n47ujf7tavu2rv5mjnx3y

Learning to rank using 1-norm regularization and convex hull reduction

Xiaofei Nan, Yixin Chen, Xin Dang, Dawn Wilkins
2010 Proceedings of the 48th Annual Southeast Regional Conference on - ACM SE '10  
However this approach increases the problem complexity from 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.  ... 
doi:10.1145/1900008.1900052 dblp:conf/ACMse/NanCDW10 fatcat:tzkeynse5jg2bkebyc7osgycdq

Human-machine interaction for real-time linear optimization

Simon Hamel, Jonathan Gaudreault, Claude-Guy Quimper, Mathieu Bouchard, Philippe Marier
2012 2012 IEEE International Conference on Systems, Man, and Cybernetics (SMC)  
Therefore, we propose a system providing the decision-makers with a 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.  ... 
doi:10.1109/icsmc.2012.6377804 dblp:conf/smc/HamelGQBM12 fatcat:arzkzixr6zcpbf4cnkytgiluwi

A New Algorithm for 3D Reconstruction from Support Functions

R.J. Gardner, M. Kiderlen
2009 IEEE Transactions on Pattern Analysis and Machine Intelligence  
In addition we offer a 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.  ... 
doi:10.1109/tpami.2008.190 pmid:19147881 fatcat:6kj3suop6ng2rf4zm2qjbidcgq

Support vector machines

Kristin P. Bennett, Colin Campbell
2000 SIGKDD Explorations  
The classification problem is used to investigate the basic concepts behind SVMs 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.  ... 
doi:10.1145/380995.380999 fatcat:mwuco6xjejhrjmcj7o4clqy6da

Computing convex hulls and counting integer points with polymake

Benjamin Assarf, Ewgenij Gawrilow, Katrin Herr, Michael Joswig, Benjamin Lorenz, Andreas Paffenholz, Thomas Rehn
2016 Mathematical Programming Computation  
The main purpose of this paper is to report on the state of the art of computing integer 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.  ... 
doi:10.1007/s12532-016-0104-z fatcat:qk3iih53zfc3fdscu54lh6wwey

Geometric Reasoning with polymake [article]

Ewgenij Gawrilow, Michael Joswig
2005 arXiv   pre-print
The mathematical software system polymake provides a wide range of functions for 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.  ... 
arXiv:math/0507273v1 fatcat:tvougotxu5a73maybpwesr4p3q

Sequential convexification in reverse convex and disjunctive programming

Egon Balas, Joseph M. Tama, Jørgen Tind
1989 Mathematical programming  
each time the 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.  ... 
doi:10.1007/bf01587096 fatcat:ezvkwghrt5dl7gmi4wzfpijeiu

Star splaying

Richard Shewchuk
2005 Proceedings of the twenty-first annual symposium on Computational geometry - SCG '05  
Star splaying is a general-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  ... 
doi:10.1145/1064092.1064129 dblp:conf/compgeom/Shewchuk05 fatcat:yopblutnv5acjfgvun26cwvnq4

Likelihood-based Inference for Exponential-Family Random Graph Models via Linear Programming [article]

Pavel N. Krivitsky
2022 arXiv   pre-print
While the 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 .  ... 
arXiv:2202.03572v1 fatcat:hjzvdg5icjdurf45piy76gitvq

On Functional Separately Convex Hulls

J. Matoušek, P. Plecháč
1998 Discrete & Computational Geometry  
Zhang on the existence of higher-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.  ... 
doi:10.1007/pl00009331 fatcat:gwcdua7ce5b5pdlqt6sx7fxoy4

Convex Hulls in a 3-Dimensional Space [chapter]

Vladimir Kovalevsky, Henrik Schulz
2004 Lecture Notes in Computer Science  
This paper describes a new algorithm of computing the 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.  ... 
doi:10.1007/978-3-540-30503-3_14 fatcat:22uhww5renh6tk3lnsx3ef2aaq
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