Given n points in ℝd and a maximum allowed tolerance e > 0, the minimum hyperplanes clustering problem consists in finding a minimum number of hyperplanes such that the Euclidean distance between each point and the nearest hyperplane is at most e. We present a column generation approach for this problem based on a mixed integer nonlinear formulation in which the master is a set covering problem and the pricing subproblem is a mixed integer program with a nonconvex normalization constraint. We propose different ways of generating the initial pool of columns and investigate their impact on the overall algorithm. Since the pricing subproblem is substantially complicated by the l2-norm constraint, we consider approximate pricing subproblems involving different norms. Some strategies for refining the solution and speeding-up the overall method are also discussed. The performance of our column generation algorithm is assessed on realistic randomly generated instances as well as on real-world instances.
{"title":"Column Generation for the Minimum Hyperplanes Clustering Problem","authors":"E. Amaldi, A. Ceselli, Kanika Dhyani","doi":"10.1287/ijoc.1120.0513","DOIUrl":"https://doi.org/10.1287/ijoc.1120.0513","url":null,"abstract":"Given n points in ℝd and a maximum allowed tolerance e > 0, the minimum hyperplanes clustering problem consists in finding a minimum number of hyperplanes such that the Euclidean distance between each point and the nearest hyperplane is at most e. We present a column generation approach for this problem based on a mixed integer nonlinear formulation in which the master is a set covering problem and the pricing subproblem is a mixed integer program with a nonconvex normalization constraint. We propose different ways of generating the initial pool of columns and investigate their impact on the overall algorithm. Since the pricing subproblem is substantially complicated by the l2-norm constraint, we consider approximate pricing subproblems involving different norms. Some strategies for refining the solution and speeding-up the overall method are also discussed. The performance of our column generation algorithm is assessed on realistic randomly generated instances as well as on real-world instances.","PeriodicalId":386200,"journal":{"name":"Cologne Twente Workshop on Graphs and Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128268343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2008-06-27DOI: 10.1007/978-3-540-69733-6_62
E. Hoshino, C. C. Souza
{"title":"Column Generation Algorithms for the Capacitated m-Ring-Star Problem","authors":"E. Hoshino, C. C. Souza","doi":"10.1007/978-3-540-69733-6_62","DOIUrl":"https://doi.org/10.1007/978-3-540-69733-6_62","url":null,"abstract":"","PeriodicalId":386200,"journal":{"name":"Cologne Twente Workshop on Graphs and Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2008-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116223057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Grigoriev, Joyce van Loon, M. Sviridenko, M. Uetz, T. Vredeveld
The tone production system for an electronic musical instrument comprises depressed key detection means for outputting pulses at a timing corresponding to a key depressed by time division timings assigned to each note, a shift register for shifting successively said pulses outputted from said depressed key detecting means in synchronism with said time division timing, key data forming means for taking out each output pulse of predetermined plural stages of said shift register in accordance with a desired type of chord, musical tone production means for producing each of plural musical tone signals of notes corresponding to pulse generation timings in accordance with each pulse taken out by said key data forming means.
{"title":"Optimal bundle pricing for homogeneous items","authors":"A. Grigoriev, Joyce van Loon, M. Sviridenko, M. Uetz, T. Vredeveld","doi":"10.26481/umamet.2006050","DOIUrl":"https://doi.org/10.26481/umamet.2006050","url":null,"abstract":"The tone production system for an electronic musical instrument comprises depressed key detection means for outputting pulses at a timing corresponding to a key depressed by time division timings assigned to each note, a shift register for shifting successively said pulses outputted from said depressed key detecting means in synchronism with said time division timing, key data forming means for taking out each output pulse of predetermined plural stages of said shift register in accordance with a desired type of chord, musical tone production means for producing each of plural musical tone signals of notes corresponding to pulse generation timings in accordance with each pulse taken out by said key data forming means.","PeriodicalId":386200,"journal":{"name":"Cologne Twente Workshop on Graphs and Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129439411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1007/978-88-7642-475-5_61
Mirjam Friesen, D. Theis
{"title":"Fooling-sets and rank in nonzero characteristic","authors":"Mirjam Friesen, D. Theis","doi":"10.1007/978-88-7642-475-5_61","DOIUrl":"https://doi.org/10.1007/978-88-7642-475-5_61","url":null,"abstract":"","PeriodicalId":386200,"journal":{"name":"Cologne Twente Workshop on Graphs and Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129032745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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