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A Friendly Smoothed Analysis of the Simplex Method [article]

Daniel Dadush, Sophie Huiberts
2019 arXiv   pre-print
One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng (JACM '04), who developed the notion of smoothed analysis.  ...  Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years.  ...  The first such method, and the main one used in the context of smoothed analysis, is the parametric objective method of Gass and Saaty [GS55] , dubbed the shadow (vertex) simplex method by Borgwardt  ... 
arXiv:1711.05667v4 fatcat:6k7ww7bodbffxkhsxrrabkxz4a

A friendly smoothed analysis of the simplex method

Daniel Dadush, Sophie Huiberts
2018 Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2018  
One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng (JACM '04), who developed the notion of smoothed analysis.  ...  Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years.  ...  The first such method, and the main one used in the context of smoothed analysis, is the parametric objective method of Gass and Saaty [35] , dubbed the shadow (vertex) simplex method by Borgwardt [14  ... 
doi:10.1145/3188745.3188826 dblp:conf/stoc/DadushH18 fatcat:d2g3rg3zcrdlxnlkhvhbwz5svi

A Friendly Smoothed Analysis of the Simplex Method

Daniel Dadush, Sophie Huiberts
2020 SIAM journal on computing (Print)  
The first such method, and the main one used in the context of smoothed analysis, is the parametric objective method of Gass and Saaty [44] , dubbed the shadow (vertex) simplex method by Borgwardt [16]  ...  A more general method, popular in the context of average case analysis, is the selfdual parametric simplex method of Dantzig [31] .  ...  The authors would like to thank the Simons Institute program on``Bridging Continuous and Discrete Optimization,"" where some of this work was done.  ... 
doi:10.1137/18m1197205 fatcat:gbnr52kzwnam5haonc7z7co5mi

Smoothed analysis of algorithms [article]

Daniel A. Spielman, Shang-Hua Teng
2002 arXiv   pre-print
Spielman and Teng introduced the smoothed analysis of algorithms to provide a framework in which one could explain the success in practice of algorithms and heuristics that could not be understood through  ...  In this talk, we survey some of the smoothed analyses that have been performed.  ...  In our paper introducing smoothed analysis, we proved that the simplex method has polynomial smoothed complexity [27] .  ... 
arXiv:math/0212413v1 fatcat:eqfdmfz2yzhi5pwaz7mgz3h5ze

Detection and Remediation of Stagnation in the Nelder--Mead Algorithm Using a Sufficient Decrease Condition

C. T. Kelley
1999 SIAM Journal on Optimization  
We also give results that apply when the objective function is a lowamplitude perturbation of a smooth function. We illustrate our results with some numerical examples.  ...  is smooth.  ...  The author is grateful to Carl Meyer and Margaret Wright for their assistance with several aspects of this work.  ... 
doi:10.1137/s1052623497315203 fatcat:yldulfbxh5ci3jriu4yi3rrwa4

Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms [article]

Roman Vershynin
2007 arXiv   pre-print
The simplex method in Linear Programming motivates several problems of asymptotic convex geometry.  ...  We discuss some conjectures and known results in two related directions -- computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses  ...  For one reason, random matrices sometimes provide an intuition for what to expect in practice; we saw such reasoning about average analysis and smoothed analysis of the simplex method in the previous section  ... 
arXiv:cs/0703093v1 fatcat:t2gw3yqgh5c7xplnam6dsp3rfe

Smoothed Analysis: Analysis of Algorithms Beyond Worst Case

Bodo Manthey, Heiko Röglin
2011 it - Information Technology  
We give a gentle, not too formal introduction to smoothed analysis by means of two examples: the k-means method for clustering and the Nemhauser/Ullmann algorithm for the knapsack problem.  ...  In order to provide a more realistic performance measure that can explain the practical performance of algorithms, smoothed analysis has been introduced.  ...  Smoothed Analysis To overcome the drawbacks of both average-case and worst-case analysis and in order to explain the speed of the simplex method, Daniel A.  ... 
doi:10.1524/itit.2011.0654 fatcat:fkiqutsij5ao7idaz6t77qepdu

A unified subdivision approach for multi-dimensional non-manifold modeling

Yu-Sung Chang, Hong Qin
2006 Computer-Aided Design  
We briefly describe the subdivision matrix analysis to ensure a reasonable smoothness across extraordinary topologies, and empirical results support our assumption.  ...  The scheme is derived from the double (k + 1)-directional box splines over k-simplicial domains. Thus, it guarantees a certain level of smoothness in the limit on a regular mesh.  ...  Acknowledgments The authors wish to thank Dr. Kevin T. McDonnell for his positive suggestions and for proof-reading the paper.  ... 
doi:10.1016/j.cad.2006.04.004 fatcat:dl6zpbl4xrh2rlwpooctdxazqy

The Viro Method for Construction of Piecewise Algebraic Hypersurfaces

Yisheng Lai, Weiping Du, Renhong Wang
2013 Abstract and Applied Analysis  
The method is based on the smooth blending theory and the Viro method for construction of Bernstein-Bézier algebraic hypersurface piece on a simplex.  ...  We propose a new method to construct a real piecewise algebraic hypersurface of a given degree with a prescribed smoothness and topology.  ...  The method is primarily based on the work of Lai et al. in [18] and the smooth blending theory.  ... 
doi:10.1155/2013/690341 fatcat:ues76egdo5g7tcqzcbl7ojl7da

Label Space: A Coupled Multi-shape Representation [chapter]

James Malcolm, Yogesh Rathi, Martha E. Shenton, Allen Tannenbaum
2008 Lecture Notes in Computer Science  
Under this framework, object labels are mapped to vertices of a regular simplex, e.g. the unit interval for two labels, a triangle for three labels, a tetrahedron for four labels, etc.  ...  This forms the basis of a convex linear structure with the property that all labels are equally spaced.  ...  It was also supported by a Marie Curie Grant through the Technion, Israel Institute of Technology. This work is part of NAMIC, funded by the NIH Roadmap for Medical Research, Grant U54 EB005149.  ... 
doi:10.1007/978-3-540-85990-1_50 fatcat:nv26gw3lube2zkxckb7h3ga53e

The Simplex Gradient and Noisy Optimization Problems [chapter]

D. M. Bortz, C. T. Kelley
1998 Computational Methods for Optimal Design and Control  
The performance of these methods can be explained in terms of the di erence approximation of the gradient implicit in the function evaluations.  ...  The Nelder-Mead, multidirectional search, and implicit ltering methods are three such methods.  ...  In this paper we show how use of that gradient information can unify, extend, and simplify the analysis of these methods in the context of this important class of problems.  ... 
doi:10.1007/978-1-4612-1780-0_5 fatcat:44z2nd45zvgmxpi7pceoh5qk4q

Tissue Reconstruction Based on Deformation of Dual Simplex Meshes [chapter]

David Svoboda, Pavel Matula
2003 Lecture Notes in Computer Science  
A new semiautomatic method for tissue reconstruction based on deformation of a dual simplex mesh was developed. The method is suitable for specifically-shaped objects.  ...  The searching procedure is based on careful analysis of object boundaries and on the assumption that the analyzed objects are sphere-like shaped.  ...  MSM-143300002) and by the Academy of Sciences of the Czech Republic (Grants No. S5004010 and No. B5004102).  ... 
doi:10.1007/978-3-540-39966-7_49 fatcat:txynwad6b5fmnnmqbsrz4fu5ti

Analysis of Cortical Shape in Children with Simplex Autism

D. L. Dierker, E. Feczko, J. R. Pruett, S. E. Petersen, B. L. Schlaggar, J. N. Constantino, J. W. Harwell, T. S. Coalson, D. C. Van Essen
2013 Cerebral Cortex  
of regional differences that are below statistical significance when using coordinate-based analysis methods.  ...  Comparisons of average midthickness surfaces of children with simplex autism and those of typical children suggest that the significant sulcal depth differences represent local peaks in a larger pattern  ...  The authors thank Erin Reid for landmark editing and surface evaluation; Sarah Hoertel for coordinating recruitment, scheduling, and assessments of the children; Kelly McVey, Katie Ihnen, Rebecca Coalson  ... 
doi:10.1093/cercor/bht294 pmid:24165833 pmcid:PMC4366616 fatcat:bvm75pn44ndtbhn34uhqsxrqgm

Cementation of a Polyethylene Liner Into a Metal Acetabular Shell

Aaron A. Hofmann, Edward J. Prince, F. Thurston Drake, Kenneth J. Hunt
2009 Journal of Arthroplasty  
(This multiplicity analysis included the Simplex smooth liners.)  ...  If the Simplex smooth liners are discarded from the Mean Failure Load on Lever-out Testing (Newtons) Standard Locking Simplex Smooth Simplex Roughened comparison (due to the testing malfunction), there  ... 
doi:10.1016/j.arth.2008.05.027 pmid:18701253 fatcat:3istcwzjgrgt5pnpxkirlpmz34

RSCS: A Parallel Simplex Algorithm for the Nimrod/O Optimization Toolset

Andrew Lewis, David Abramson, Tom Peachey
2006 Scientific Programming  
This paper describes a method of parallelisation of the popular Nelder-Mead simplex optimization algorithms that can lead to enhanced performance on parallel and distributed computing resources.  ...  A reducing set of simplex vertices are used to derive search directions generally closely aligned with the local gradient.  ...  $jobname endtask method simplex starts 8 named "simplex" starting points random tolerance 0.005 endstarts endmethod The plan file starts by defining the parameters.  ... 
doi:10.1155/2006/906394 fatcat:5cgwzffyfvb4ld5dd42urzdvvm
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