An illumination problem with tradeoff between coverage of a dataset and aperture angle of a conic light beam

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• 作者：Annabella Astorino ; Manlio Gaudioso ; Alberto Seeger
• 关键词：Illumination problem ; Conic light beam ; Aperture angle ; Sharpness coefficient ; Nonsmooth optimization
• 刊名：Optimization and Engineering
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：17
• 期：3
• 页码：557-575
• 全文大小：769 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Optimization
  Engineering, general
  Systems Theory and Control
  Environmental Management
  Agriculture
  Operation Research and Decision Theory
  
• 出版者：Springer Netherlands
• ISSN：1573-2924
• 卷排序：17

文摘

Let \(\{a_i:i\in I\}\) be a finite set in \({\mathbb{R}}^n\). The illumination problem addressed in this work concerns the optimal location and orientation of a conic light beam $$ R\big (z,y,s\big )= \left\{ x \in {\mathbb{R}}^n : s\,\Vert x-z\Vert - \langle y, x-z\rangle \le 0\right\} .$$The aperture angle \(\vartheta = 2\arccos s\) of the conic light beam is a decreasing function of the sharpness coefficient \( s\in [0,1]\). The problem at hand is to select an apex z in a prescribed compact region \(Z\subseteq {\mathbb{R}}^n\) and a unit vector \(y\in {\mathbb{R}}^n\) so that the conic light beam R(z, y, s) fulfils two conflicting requirements: it captures as many points \(a_i\) as possible and, at the same time, it has a sharpness coefficient s as large as possible.KeywordsIllumination problemConic light beamAperture angleSharpness coefficientNonsmooth optimizationMathematics Subject Classification90C2590C2690C40ReferencesAstorino A, Gaudioso M, Seeger A (2014) An illumination problem: optimal apex and optimal orientation for a cone of light. J Glob Optim 58:729–750MathSciNetCrossRefMATHGoogle ScholarAstorino A, Gaudioso M, Seeger A (2015) Central axes and peripheral points in high dimensional directional datasets. Comput Optim Appl. doi:10.1007/s10589-014-9724-2 Bose P, Hurtado-Diaz F, Omana-Pulido E, Snoeyink J, Toussaint GT (2002) Some aperture-angle optimization problems. Algorithmica 33:411–435MathSciNetCrossRefMATHGoogle ScholarClarke FH (1975) Generalized gradients and applications. Trans Am Math Soc 205:247–262MathSciNetCrossRefMATHGoogle ScholarCorrea R, Seeger A (1985) Directional derivative of a minimax function. Nonlinear Anal 9:13–22MathSciNetCrossRefMATHGoogle ScholarDanskin JM (1966) The theory of max-min, with applications. SIAM J Appl Math 14:641–664MathSciNetCrossRefMATHGoogle ScholarDunkl CF, Williams KS (1964) A simple norm inequality. Am Math Mon 71:53–54MathSciNetCrossRefMATHGoogle ScholarFuduli A, Gaudioso M, Giallombardo G (2004) Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J Optim 14:743–756MathSciNetCrossRefMATHGoogle ScholarHiriart-Urruty J-B, Lemaréchal C (1993) Convex analysis and minimization algorithms, I–II. Springer-Verlag, BerlinMATHGoogle ScholarOmana-Pulido E, Toussaint GT (2002) Aperture-angle optimization problems in three dimensions. J Math Model Algorithms 1:301–329MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer Science+Business Media New York 2015Authors and AffiliationsAnnabella Astorino1Manlio Gaudioso2Email authorAlberto Seeger31.Istituto di Calcolo e Reti ad Alte Prestazioni C.N.R., c/o Dipartimento di Elettronica Informatica e SistemisticaUniversità delle CalabriaRendeItaly2.Dipartimento di Elettronica Informatica e SistemisticaUniversità delle CalabriaRendeItaly3.Department of MathematicsUniversity of AvignonAvignonFrance About this article CrossMark Print ISSN 1389-4420 Online ISSN 1573-2924 Publisher Name Springer US About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s11081-015-9293-y_An illumination problem with trade", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s11081-015-9293-y_An illumination problem with trade", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips