Finding largest small polygons with GloptiPoly
A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n . Many instances are already solved in the literature, namely for all odd n , and for n = 4, 6 and 8. Thus, for even n ≥ 10, instances of this prob...
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| Published in: | Journal of global optimization Vol. 56; no. 3; pp. 1017 - 1028 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01-07-2013
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| Abstract | A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices
n
. Many instances are already solved in the literature, namely for all odd
n
, and for
n
= 4, 6 and 8. Thus, for even
n
≥ 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for
n
= 10 and
n
= 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic. |
|---|---|
| AbstractList | A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and $8$. Thus, for even $n\geq 10$, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for $n=10$ and $n=12$. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic. Issue Title: Special Issue: Toulouse Global OptimizationWorkshop 2010 A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n >= 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic.[PUBLICATION ABSTRACT] A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n . Many instances are already solved in the literature, namely for all odd n , and for n = 4, 6 and 8. Thus, for even n ≥ 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic. A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n greater than or equal to 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic. |
| Author | Henrion, Didier Messine, Frédéric |
| Author_xml | – sequence: 1 givenname: Didier surname: Henrion fullname: Henrion, Didier organization: CNRS, LAAS, Université de Toulouse, UPS, INSA, INP, ISAE, UT1, UTM, LAAS, Faculty of Electrical Engineering, Czech Technical University in Prague – sequence: 2 givenname: Frédéric surname: Messine fullname: Messine, Frédéric email: messine@n7.fr organization: ENSEEIHT-IRIT, Université de Toulouse |
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| Cites_doi | 10.1016/j.jcta.2007.02.006 10.1080/10556780802699201 10.1142/p665 10.1145/779359.779363 10.1007/s00454-006-1238-y 10.1007/s10898-006-9065-5 10.1016/j.ejor.2009.01.025 10.1016/0097-3165(75)90004-7 10.1006/jcta.2001.3225 10.1007/978-1-4614-0769-0_20 10.1090/fic/055/01 10.1007/s101079900106 10.1137/05064504X |
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| Copyright | Springer Science+Business Media, LLC. 2011 Springer Science+Business Media New York 2013 Distributed under a Creative Commons Attribution 4.0 International License |
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| Issue | 3 |
| Keywords | Extremal convex polygons Global optimization Semidefinite programming Nonconvex quadratic programming |
| Language | English |
| License | Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 |
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| PublicationSubtitle | An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering |
| PublicationTitle | Journal of global optimization |
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| References | de Klerk (CR6) 2010; 201 Audet, Hansen, Messine (CR2) 2007; 38 Audet, Hansen, Messine, Pardalos, Coleman (CR3) 2009 Graham (CR9) 1975; 18 Mossinghoff (CR18) 2006; 36 CR15 Reinhardt (CR19) 1922; 31 Bieri (CR5) 1961; 16 Foster, Szabo (CR8) 2007; 114 CR13 Woodall, Welsh (CR20) 1971 CR12 Lasserre (CR17) 2009 CR21 Lasserre (CR16) 2000; 331 Floudas, Pardalos (CR7) 2004 Henrion, Lasserre, Löfberg (CR11) 2009; 24 Henrion, Lasserre (CR10) 2003; 29 Audet, Hansen, Jaumard, Savard (CR1) 2000; 87 Jansson (CR14) 2006 Audet, Hansen, Messine, Xiong (CR4) 2002; 98 J.B. Lasserre (9818_CR17) 2009 E. Klerk de (9818_CR6) 2010; 201 C. Audet (9818_CR3) 2009 C. Audet (9818_CR4) 2002; 98 J. Foster (9818_CR8) 2007; 114 H. Bieri (9818_CR5) 1961; 16 9818_CR12 C. Audet (9818_CR2) 2007; 38 9818_CR13 K. Reinhardt (9818_CR19) 1922; 31 D. Henrion (9818_CR10) 2003; 29 D. Henrion (9818_CR11) 2009; 24 9818_CR15 D.R. Woodall (9818_CR20) 1971 R.L. Graham (9818_CR9) 1975; 18 M.J. Mossinghoff (9818_CR18) 2006; 36 C. Audet (9818_CR1) 2000; 87 (9818_CR7) 2004 J.B. Lasserre (9818_CR16) 2000; 331 9818_CR21 C. Jansson (9818_CR14) 2006 |
| References_xml | – ident: CR21 – volume: 114 start-page: 1515 issue: 8 year: 2007 end-page: 1525 ident: CR8 article-title: Diameter graphs of polygons and the proof of a conjecture of Graham publication-title: J. Combin. Theory Series A doi: 10.1016/j.jcta.2007.02.006 contributor: fullname: Szabo – volume: 24 start-page: 761 issue: 4–5 year: 2009 end-page: 779 ident: CR11 article-title: GloptiPoly 3: moments, optimization and semidefinite programming publication-title: Optim. Methods Softw. doi: 10.1080/10556780802699201 contributor: fullname: Löfberg – ident: CR15 – ident: CR12 – year: 2009 ident: CR17 publication-title: Moments, Positive Polynomials and Their Applications doi: 10.1142/p665 contributor: fullname: Lasserre – volume: 29 start-page: 165 issue: 2 year: 2003 end-page: 194 ident: CR10 article-title: GloptiPoly: global optimization over polynomials with Matlab and SeDuMi publication-title: ACM Trans. Math. Softw. doi: 10.1145/779359.779363 contributor: fullname: Lasserre – volume: 36 start-page: 363 year: 2006 end-page: 379 ident: CR18 article-title: Isodiametric problems for polygons publication-title: Discret. Computat. Geom. doi: 10.1007/s00454-006-1238-y contributor: fullname: Mossinghoff – ident: CR13 – volume: 16 start-page: 105 year: 1961 end-page: 106 ident: CR5 article-title: Ungelöste Probleme: Zweiter Nachtrag zu Nr. 12 publication-title: Elemente der Mathematik contributor: fullname: Bieri – year: 2006 ident: CR14 publication-title: VSDP: A MATLAB software package for Verified Semidefinite Programming International Symposium on Nonlinear Theory and its Applications (NOLTA) contributor: fullname: Jansson – volume: 87 start-page: 131 issue: 1, Series A year: 2000 end-page: 152 ident: CR1 article-title: A branch and cut algorithm for nonconvex quadratically constrained quadratic programming publication-title: Math. Program. contributor: fullname: Savard – year: 1971 ident: CR20 article-title: Thrackles and deadlock publication-title: Combinatorial Mathematics and Its Applications contributor: fullname: Welsh – volume: 38 start-page: 163 year: 2007 end-page: 179 ident: CR2 article-title: Extremal problems for convex polygons publication-title: J. Glob. Optim. doi: 10.1007/s10898-006-9065-5 contributor: fullname: Messine – volume: 31 start-page: 251 year: 1922 end-page: 270 ident: CR19 article-title: Extremale polygone gegebenen durchmessers publication-title: Jahresbericht der deutschen Mathematiker Vereinigung contributor: fullname: Reinhardt – year: 2004 ident: CR7 publication-title: Frontiers in Global Optimization—Nonconvex optimization and its applications, vol. 74 contributor: fullname: Pardalos – volume: 331 start-page: 929 issue: Série 1 year: 2000 end-page: 934 ident: CR16 article-title: Optimisation globale et théorie des moments publication-title: Comptes Rendus de l’Académie des Sciences, Paris contributor: fullname: Lasserre – volume: 201 start-page: 1 issue: 1 year: 2010 end-page: 10 ident: CR6 article-title: Exploiting special structure in semidefinite programming: a survey of theory and applications publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2009.01.025 contributor: fullname: de Klerk – volume: 18 start-page: 165 year: 1975 end-page: 170 ident: CR9 article-title: The largest small hexagon publication-title: J. Combin. Theory Series A doi: 10.1016/0097-3165(75)90004-7 contributor: fullname: Graham – volume: 98 start-page: 46 issue: 1 year: 2002 end-page: 59 ident: CR4 article-title: The largest small octagon publication-title: J. Comb. Theory Series A doi: 10.1006/jcta.2001.3225 contributor: fullname: Xiong – start-page: 1 year: 2009 end-page: 16 ident: CR3 article-title: Extremal problems for convex polygons: an update publication-title: Lectures on Global Optimization contributor: fullname: Coleman – ident: 9818_CR13 doi: 10.1007/978-1-4614-0769-0_20 – volume: 31 start-page: 251 year: 1922 ident: 9818_CR19 publication-title: Jahresbericht der deutschen Mathematiker Vereinigung contributor: fullname: K. Reinhardt – volume: 201 start-page: 1 issue: 1 year: 2010 ident: 9818_CR6 publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2009.01.025 contributor: fullname: E. Klerk de – start-page: 1 volume-title: Lectures on Global Optimization year: 2009 ident: 9818_CR3 doi: 10.1090/fic/055/01 contributor: fullname: C. Audet – ident: 9818_CR12 – volume: 98 start-page: 46 issue: 1 year: 2002 ident: 9818_CR4 publication-title: J. Comb. Theory Series A doi: 10.1006/jcta.2001.3225 contributor: fullname: C. Audet – volume: 18 start-page: 165 year: 1975 ident: 9818_CR9 publication-title: J. Combin. Theory Series A doi: 10.1016/0097-3165(75)90004-7 contributor: fullname: R.L. Graham – volume: 29 start-page: 165 issue: 2 year: 2003 ident: 9818_CR10 publication-title: ACM Trans. Math. Softw. doi: 10.1145/779359.779363 contributor: fullname: D. Henrion – volume: 87 start-page: 131 issue: 1, Series A year: 2000 ident: 9818_CR1 publication-title: Math. Program. doi: 10.1007/s101079900106 contributor: fullname: C. Audet – volume: 331 start-page: 929 issue: Série 1 year: 2000 ident: 9818_CR16 publication-title: Comptes Rendus de l’Académie des Sciences, Paris contributor: fullname: J.B. Lasserre – volume-title: Frontiers in Global Optimization—Nonconvex optimization and its applications, vol. 74 year: 2004 ident: 9818_CR7 – ident: 9818_CR15 doi: 10.1137/05064504X – volume: 16 start-page: 105 year: 1961 ident: 9818_CR5 publication-title: Elemente der Mathematik contributor: fullname: H. Bieri – ident: 9818_CR21 – volume: 36 start-page: 363 year: 2006 ident: 9818_CR18 publication-title: Discret. Computat. Geom. doi: 10.1007/s00454-006-1238-y contributor: fullname: M.J. Mossinghoff – volume-title: Combinatorial Mathematics and Its Applications year: 1971 ident: 9818_CR20 contributor: fullname: D.R. Woodall – volume: 38 start-page: 163 year: 2007 ident: 9818_CR2 publication-title: J. Glob. Optim. doi: 10.1007/s10898-006-9065-5 contributor: fullname: C. Audet – volume: 114 start-page: 1515 issue: 8 year: 2007 ident: 9818_CR8 publication-title: J. Combin. Theory Series A doi: 10.1016/j.jcta.2007.02.006 contributor: fullname: J. Foster – volume-title: VSDP: A MATLAB software package for Verified Semidefinite Programming International Symposium on Nonlinear Theory and its Applications (NOLTA) year: 2006 ident: 9818_CR14 contributor: fullname: C. Jansson – volume: 24 start-page: 761 issue: 4–5 year: 2009 ident: 9818_CR11 publication-title: Optim. Methods Softw. doi: 10.1080/10556780802699201 contributor: fullname: D. Henrion – volume-title: Moments, Positive Polynomials and Their Applications year: 2009 ident: 9818_CR17 doi: 10.1142/p665 contributor: fullname: J.B. Lasserre |
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| Snippet | A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices
n
. Many... Issue Title: Special Issue: Toulouse Global OptimizationWorkshop 2010 A small polygon is a convex polygon of unit diameter. We are interested in small polygons... A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many... A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many... |
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| SubjectTerms | Computer Science Conics Mathematical models Mathematics Mathematics and Statistics Matlab Operations Research/Decision Theory Optimization Optimization algorithms Optimization and Control Packages Polygons Polynomials Quadratic programming Real Functions Semidefinite programming Solvers Studies Symmetry |
| Title | Finding largest small polygons with GloptiPoly |
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