{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T10:53:23Z","timestamp":1782384803143,"version":"3.54.5"},"reference-count":17,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","funder":[{"name":"Beijing Natural Science Foundation Project","award":["Z200002"],"award-info":[{"award-number":["Z200002"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12131003"],"award-info":[{"award-number":["12131003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11201333"],"award-info":[{"award-number":["11201333"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11901558"],"award-info":[{"award-number":["11901558"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11871081"],"award-info":[{"award-number":["11871081"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asia Pac. J. Oper. Res."],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:p> In this paper, we consider the spherical [Formula: see text]-means problem with penalties, a robust model of spherical clusterings that requires identifying outliers during clustering to improve the quality of the solution. Each outlier will incur a specified penalty cost. In this problem, one should detect the outliers and propose a [Formula: see text]-clustering for the given data set so as to minimize the sum of the clustering and penalty costs. As our main contribution, we present a [Formula: see text]-approximation via single-swap local search and an [Formula: see text]-approximation via multi-swap local search. <\/jats:p>","DOI":"10.1142\/s0217595922400140","type":"journal-article","created":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T04:52:47Z","timestamp":1670993567000},"source":"Crossref","is-referenced-by-count":3,"title":["Approximation Algorithms for Spherical k-Means Problem with Penalties Using Local Search Techniques"],"prefix":"10.1142","volume":"40","author":[{"given":"Xiaoyun","family":"Tian","sequence":"first","affiliation":[{"name":"Department of Operations, Research and Information Engineering, Beijing University of Technology, Beijing 100124, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ling","family":"Gai","sequence":"additional","affiliation":[{"name":"Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yicheng","family":"Xu","sequence":"additional","affiliation":[{"name":"Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongmei","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250101, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2023,1,4]]},"reference":[{"key":"S0217595922400140BIB001","first-page":"1027","volume-title":"Proc. ACM-SIAM Symp. Discrete Algorithms","author":"Arthur D","year":"2007"},{"issue":"1","key":"S0217595922400140BIB002","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1023\/A:1007612920971","volume":"42","author":"Dhillon I","year":"2001","journal-title":"Machine Learning"},{"key":"S0217595922400140BIB003","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/978-3-319-23240-9_9","volume-title":"Proc. Modeling Decisions for Artificial Intelligence","author":"Endo Y","year":"2015"},{"key":"S0217595922400140BIB004","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1007\/978-3-030-18126-0_15","volume-title":"Proc. Int. Workshop on Frontiers in Algorithmics","author":"Feng Q","year":"2019"},{"issue":"10","key":"S0217595922400140BIB005","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v050.i10","volume":"50","author":"Hornik K","year":"2012","journal-title":"Journal of Statistical Software"},{"key":"S0217595922400140BIB006","doi-asserted-by":"crossref","first-page":"1977","DOI":"10.1007\/s10878-020-00569-1","volume":"44","author":"Ji S","year":"2022","journal-title":"Journal of Combinatorial Optimization"},{"issue":"2","key":"S0217595922400140BIB007","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.comgeo.2004.03.003","volume":"28","author":"Kanungo T","year":"2004","journal-title":"Computational Geometry"},{"issue":"4","key":"S0217595922400140BIB008","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1007\/s10898-019-00779-w","volume":"76","author":"Li M","year":"2020","journal-title":"Journal of Global Optimization"},{"issue":"2","key":"S0217595922400140BIB009","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1109\/TIT.1982.1056489","volume":"28","author":"Lloyd S","year":"1982","journal-title":"IEEE Transactions on Information Theory"},{"key":"S0217595922400140BIB010","first-page":"281","volume-title":"Proc. Berkeley Symp. Mathematical Statistics and Probability, Statistics","author":"Macqueen J","year":"1966"},{"key":"S0217595922400140BIB011","volume-title":"Introduction to Modern Retrieval.","author":"Salton G","year":"1983"},{"issue":"4","key":"S0217595922400140BIB013","doi-asserted-by":"crossref","first-page":"2277","DOI":"10.3934\/jimo.2021067","volume":"18","author":"Wu C","year":"2022","journal-title":"Journal of Industrial and Management Optimization"},{"issue":"1","key":"S0217595922400140BIB014","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s10898-015-0394-0","volume":"67","author":"Xu Y","year":"2017","journal-title":"Journal of Global Optimization"},{"key":"S0217595922400140BIB015","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.tcs.2017.03.014","volume":"774","author":"Xu Y","year":"2019","journal-title":"Theoretical Computer Science"},{"issue":"4","key":"S0217595922400140BIB016","first-page":"1950022:1","volume":"6","author":"Yang R","year":"2019","journal-title":"Asia-Pacific Journal of Operational Research"},{"key":"S0217595922400140BIB017","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.tcs.2020.06.029","volume":"853","author":"Zhang D","year":"2021","journal-title":"Theoretical Computer Science"},{"issue":"2","key":"S0217595922400140BIB018","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/s10878-018-0278-6","volume":"37","author":"Zhang D","year":"2019","journal-title":"Journal of Combinatorial Optimization"}],"container-title":["Asia-Pacific Journal of Operational Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0217595922400140","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,16]],"date-time":"2023-03-16T05:53:19Z","timestamp":1678945999000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0217595922400140"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,4]]},"references-count":17,"journal-issue":{"issue":"01","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["10.1142\/S0217595922400140"],"URL":"https:\/\/doi.org\/10.1142\/s0217595922400140","relation":{},"ISSN":["0217-5959","1793-7019"],"issn-type":[{"value":"0217-5959","type":"print"},{"value":"1793-7019","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,4]]},"article-number":"2240014"}}