Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to ma...

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Published in:	Journal of global optimization Vol. 49; no. 2; pp. 185 - 212
Main Authors:	Nataraj, P. S. V., Arounassalame, M.
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.02.2011 Springer Nature B.V
Subjects:	Algorithms Computer Science Mathematical programming Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Polynomials Propagation Real Functions Studies Pruning Constrained global optimization Subdivision Bernstein polynomials
ISSN:	0925-5001, 1573-2916
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Abstract	We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.
AbstractList	We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.[PUBLICATION ABSTRACT] We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.
Author	Nataraj, P. S. V. Arounassalame, M.
Author_xml	– sequence: 1 givenname: P. S. V. surname: Nataraj fullname: Nataraj, P. S. V. email: nataraj@sc.iitb.ac.in, nataraj@ee.iitb.ac.in organization: Systems and Control Engineering, Indian Institute of Technology Bombay – sequence: 2 givenname: M. surname: Arounassalame fullname: Arounassalame, M. organization: Systems and Control Engineering, Indian Institute of Technology Bombay
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CitedBy_id	crossref_primary_10_1109_TRO_2020_3036617 crossref_primary_10_1007_s13198_017_0673_x crossref_primary_10_1007_s11633_012_0619_3 crossref_primary_10_1007_s10898_012_0024_z crossref_primary_10_1007_s41066_018_0134_1 crossref_primary_10_1007_s13198_019_00922_6 crossref_primary_10_1007_s10898_015_0358_4 crossref_primary_10_1016_j_amc_2017_07_026 crossref_primary_10_1007_s11786_014_0198_5 crossref_primary_10_1016_j_cam_2020_113377 crossref_primary_10_1007_s13198_017_0639_z crossref_primary_10_1080_0305215X_2018_1521399 crossref_primary_10_1016_j_jprocont_2011_12_006 crossref_primary_10_1016_j_amc_2014_01_114 crossref_primary_10_1016_j_cagd_2012_03_001 crossref_primary_10_1016_j_amc_2018_08_049 crossref_primary_10_1007_s40314_015_0224_5
Cites_doi	10.1007/978-1-4757-3040-1 10.1007/BF01097060 10.1007/978-94-017-1247-7_7 10.1023/A:1014750702474 10.1109/9.661615 10.1007/s10898-004-5906-2 10.1016/S0377-2217(96)00310-4 10.1007/3-540-53032-0 10.1007/978-1-4615-2025-2 10.1007/978-1-4757-2495-0 10.1145/1132973.1132980 10.1023/B:REOM.0000003995.08805.2a 10.1016/j.cam.2005.08.037 10.1016/0098-1354(94)E0032-I 10.1007/978-1-4471-0249-6 10.1007/s11633-007-0342-7
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Keywords	Pruning Constrained global optimization Subdivision Bernstein polynomials
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References	GranvilliersL.BenhamouF.Algorithm 852: real paver: an interval solver using constraint satisfaction techniquesACM Trans. Math. Softw.200632113815610.1145/1132973.1132980 Hedar, A-R., Ahmed, A.: Studies on Metaheuristics for continuous global optimization problems. Ph.D. dissertation, Kyoto University, Kyoto, Japan (2004) RatzD.CsendesT.On the selection of subdivision directions in interval Branch-and-Bound methods for global optimizationJ. Global Optim.19957218320710.1007/BF01097060 JaulinL.KiefferM.DidritO.Applied Interval Analysis2001LondonSpringer Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global optimization algorithms. Springer, London (1990) LebbahY.MichelC.RueherM.An efficient and safe framework for solving optimization problemsJ. Comput. Appl. Math.200719937237710.1016/j.cam.2005.08.037 HansenE.WalsterG.W.Global Optimization Using Interval Analysis2004New YorkMarcel Dekker Inc VaidyanathanR.Ed-halwagiM.Global optimization of nonconvex nonlinear programs via interval analysisComput. Chem. Eng.1994181088989710.1016/0098-1354(94)E0032-I GranvilliersL.On the combination of interval constraint solversReliab. Comput.2001746748310.1023/A:1014750702474 Stahl. V.: Interval methods for bounding the range of polynomials and solving systems of nonlinear equations. Ph.D. dissertation, University of linz (2005) Shaswathi Ray, S.: A new approach to range computation of polynomials using the Bernstein form. Ph.D. dissertation, Indian Institute of Technology Bombay, Mumbai, India (2006) Rump, S.M.: INTLAB-Interval Laboratory, Developments in Reliable Computing. Kluwer, Dordrecht, pp. 77–104 (1999) MATLAB Reference Guide.: MATLAB version 6.1, The Mathworks Inc., Natick, MA (2001) Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.: Revising hull and box consistency. In: Proceedings of the sixteenth international conference on logic programming (ICLP’99), The MIT Press, Las Cruces, USA, pp. 230–244 (1999) Henrion, D., Lasserre J.B.: GloptiPoly: global optimization with polynomials with MATLAB and SeDuMi, version 2.3.0 (2006) VenkataramanR.Applied Optimization with MATLAB Programming2001LondonWiley NatarajP.S.V.KotechaK.An improved interval global optimization algorithm using higher order inclusion function formsJ. Global Optim.2005321356310.1007/s10898-004-5906-2 NatarajP.S.V.ArounassalameM.A new subdivision algorithm for bernstein polynomial approach to global optimizationInt. J. Automat. Comput.20074434235210.1007/s11633-007-0342-7 LiH.L.ChangC.T.An approximate approach of global optimization for polynomial programming problemsEur J Oper Res199810762563210.1016/S0377-2217(96)00310-4 GarloffJ.The Bernstein algorithmInterval Comput.19932154168 KearfottR.B.Rigorous Global Search: Continuous problems1996DordrechtKluwer GarloffJ.MunroN.Solving strict polynomial inequalities by Bernstein expansion. The use of symbolic methods in control system analysis and designIEE Control Engineering Series, vol. 561999BerlinSpringer339352 ZettlerM.GarloffJ.Robustness analysis of polynomials with polynomial parameter dependency using bernstein expansionIEEE Trans. Autom. Contr.199843542543110.1109/9.661615 FloudasC.A.PardalosP.M.Handbook of Test problems in Local and Global Optimization1999The NetherlandsKluwer GarloffJ.NickelKConvergent bounds for range of multivariate polynomials. Interval mathematicsLecture Notes in Computer Science, vol. 2121985BerlinSpringer3756 Global Library, Avaiable from http://www.gamsworld.org/global/globallib. RatschekH.RokneJ.Computer Methods for the Range of Functions1984ChichesterEllis Horwood Limited Publishers HorstR.PardalosP.M.Handbook of Global Optimization1995The NetherlandsKluwer Malan, S., Milanese, M., Tarangna, M., Garloff, J.: B3 algorithm for robust performances analysis in presence of mixed parametric and dynamic perturbations. In: Proceedings of 31st Conference on Decision and Control, Tucson, Arizona, pp. 128–133 (1992) NatarajP.S.V.KotechaK.Global optimization with higher order inclusion function forms part 1: a combined Taylor–Bernstein formReliab. Comput.2004101274410.1023/B:REOM.0000003995.08805.2a L. Jaulin (9485_CR14) 2001 H.L. Li (9485_CR17) 1998; 107 L. Granvilliers (9485_CR10) 2006; 32 J. Garloff (9485_CR6) 1993; 2 9485_CR8 L. Granvilliers (9485_CR9) 2001; 7 9485_CR26 9485_CR27 R. Vaidyanathan (9485_CR28) 1994; 18 J. Garloff (9485_CR7) 1999 P.S.V. Nataraj (9485_CR21) 2004; 10 9485_CR25 P.S.V. Nataraj (9485_CR22) 2007; 4 R. Horst (9485_CR13) 1995 R. Venkataraman (9485_CR29) 2001 R.B. Kearfott (9485_CR15) 1996 D. Ratz (9485_CR24) 1995; 7 E. Hansen (9485_CR11) 2004 J. Garloff (9485_CR5) 1985 M. Zettler (9485_CR30) 1998; 43 9485_CR3 P.S.V. Nataraj (9485_CR20) 2005; 32 9485_CR2 H. Ratschek (9485_CR23) 1984 C.A. Floudas (9485_CR4) 1999 9485_CR12 Y. Lebbah (9485_CR16) 2007; 199 9485_CR1 9485_CR19 9485_CR18
References_xml	– reference: FloudasC.A.PardalosP.M.Handbook of Test problems in Local and Global Optimization1999The NetherlandsKluwer – reference: Global Library, Avaiable from http://www.gamsworld.org/global/globallib. – reference: KearfottR.B.Rigorous Global Search: Continuous problems1996DordrechtKluwer – reference: Hedar, A-R., Ahmed, A.: Studies on Metaheuristics for continuous global optimization problems. Ph.D. dissertation, Kyoto University, Kyoto, Japan (2004) – reference: NatarajP.S.V.ArounassalameM.A new subdivision algorithm for bernstein polynomial approach to global optimizationInt. J. Automat. Comput.20074434235210.1007/s11633-007-0342-7 – reference: Stahl. V.: Interval methods for bounding the range of polynomials and solving systems of nonlinear equations. Ph.D. dissertation, University of linz (2005) – reference: Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.: Revising hull and box consistency. In: Proceedings of the sixteenth international conference on logic programming (ICLP’99), The MIT Press, Las Cruces, USA, pp. 230–244 (1999) – reference: GarloffJ.NickelKConvergent bounds for range of multivariate polynomials. Interval mathematicsLecture Notes in Computer Science, vol. 2121985BerlinSpringer3756 – reference: HorstR.PardalosP.M.Handbook of Global Optimization1995The NetherlandsKluwer – reference: VenkataramanR.Applied Optimization with MATLAB Programming2001LondonWiley – reference: Henrion, D., Lasserre J.B.: GloptiPoly: global optimization with polynomials with MATLAB and SeDuMi, version 2.3.0 (2006) – reference: LebbahY.MichelC.RueherM.An efficient and safe framework for solving optimization problemsJ. Comput. Appl. Math.200719937237710.1016/j.cam.2005.08.037 – reference: Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global optimization algorithms. Springer, London (1990) – reference: GarloffJ.MunroN.Solving strict polynomial inequalities by Bernstein expansion. The use of symbolic methods in control system analysis and designIEE Control Engineering Series, vol. 561999BerlinSpringer339352 – reference: HansenE.WalsterG.W.Global Optimization Using Interval Analysis2004New YorkMarcel Dekker Inc – reference: RatschekH.RokneJ.Computer Methods for the Range of Functions1984ChichesterEllis Horwood Limited Publishers – reference: GranvilliersL.BenhamouF.Algorithm 852: real paver: an interval solver using constraint satisfaction techniquesACM Trans. Math. Softw.200632113815610.1145/1132973.1132980 – reference: ZettlerM.GarloffJ.Robustness analysis of polynomials with polynomial parameter dependency using bernstein expansionIEEE Trans. Autom. Contr.199843542543110.1109/9.661615 – reference: MATLAB Reference Guide.: MATLAB version 6.1, The Mathworks Inc., Natick, MA (2001) – reference: NatarajP.S.V.KotechaK.An improved interval global optimization algorithm using higher order inclusion function formsJ. Global Optim.2005321356310.1007/s10898-004-5906-2 – reference: GranvilliersL.On the combination of interval constraint solversReliab. Comput.2001746748310.1023/A:1014750702474 – reference: Malan, S., Milanese, M., Tarangna, M., Garloff, J.: B3 algorithm for robust performances analysis in presence of mixed parametric and dynamic perturbations. In: Proceedings of 31st Conference on Decision and Control, Tucson, Arizona, pp. 128–133 (1992) – reference: RatzD.CsendesT.On the selection of subdivision directions in interval Branch-and-Bound methods for global optimizationJ. Global Optim.19957218320710.1007/BF01097060 – reference: Rump, S.M.: INTLAB-Interval Laboratory, Developments in Reliable Computing. Kluwer, Dordrecht, pp. 77–104 (1999) – reference: JaulinL.KiefferM.DidritO.Applied Interval Analysis2001LondonSpringer – reference: NatarajP.S.V.KotechaK.Global optimization with higher order inclusion function forms part 1: a combined Taylor–Bernstein formReliab. Comput.2004101274410.1023/B:REOM.0000003995.08805.2a – reference: LiH.L.ChangC.T.An approximate approach of global optimization for polynomial programming problemsEur J Oper Res199810762563210.1016/S0377-2217(96)00310-4 – reference: GarloffJ.The Bernstein algorithmInterval Comput.19932154168 – reference: Shaswathi Ray, S.: A new approach to range computation of polynomials using the Bernstein form. Ph.D. dissertation, Indian Institute of Technology Bombay, Mumbai, India (2006) – reference: VaidyanathanR.Ed-halwagiM.Global optimization of nonconvex nonlinear programs via interval analysisComput. Chem. 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Snippet	We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein...
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SubjectTerms	Algorithms Computer Science Mathematical programming Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Polynomials Propagation Real Functions Studies
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Title	Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm
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