Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Phân tích thứ tự hội tụ cho các ràng buộc và làm mềm dựa trên bất đẳng thức vi phân của các nghiệm của ODEs
Tóm tắt
Đối với hiệu suất của các thuật toán tối ưu toàn cục, tỷ lệ hội tụ của các sự thư giãn lồi tới các hàm mục tiêu và ràng buộc là rất quan trọng. Chúng tôi mở rộng kết quả từ Bompadre và Mitsos (J Glob Optim 52(1):1–28, 2012) để xác định tỷ lệ hội tụ của các ràng buộc tham số và sự thư giãn của các nghiệm của phương trình vi phân thường (ODEs). Những ràng buộc và sự thư giãn này được sử dụng cho tối ưu hóa động toàn cục và được tính toán thông qua các hệ ODE bổ trợ sử dụng số học khoảng và sự thư giãn McCormick. Hai phương pháp thư giãn ODE (Scott et al. trong Optim Control Appl Methods 34(2):145–163, 2013; Scott và Barton trong J Glob Optim 57:143–176, 2013) được chỉ ra là mang lại sự hội tụ bậc hai, tuy nhiên chúng có thể cư xử rất khác nhau trong thực tiễn. Khi thời gian trôi qua, hệ số trước trong ràng buộc về thứ tự hội tụ có xu hướng phát triển chậm hơn nhiều đối với một trong các phương pháp này, và thậm chí có thể giảm theo thời gian, tạo ra các quy trình tối ưu hóa toàn cục cần ít thời gian tính toán đáng kể hơn.
Từ khóa
#tối ưu hóa toàn cục #phương trình vi phân thường #hội tụ #làm mềm #bất đẳng thức vi phânTài liệu tham khảo
citation_journal_title=J. Comput. Appl. Math.; citation_title=Interval analysis: theory and applications; citation_author=G Alefeld, G Mayer; citation_volume=121; citation_issue=1–2; citation_publication_date=2000; citation_pages=421-464; citation_doi=10.1016/S0377-0427(00)00342-3; citation_id=CR1
citation_journal_title=Biotechnol. Prog.; citation_title=Stochastic dynamic optimization of batch and semicontinuous bioprocesses; citation_author=JR Banga, AA Alonso, RP Singh; citation_volume=13; citation_publication_date=1997; citation_pages=326-335; citation_doi=10.1021/bp970015+; citation_id=CR2
citation_title=Nonlinear Programming; citation_publication_date=1999; citation_id=CR3; citation_author=DP Bertsekas; citation_publisher=Athena Scientific
citation_journal_title=J. Glob. Optim.; citation_title=Convergence rate of McCormick relaxations; citation_author=A Bompadre, A Mitsos; citation_volume=52; citation_issue=1; citation_publication_date=2012; citation_pages=1-28; citation_doi=10.1007/s10898-011-9685-2; citation_id=CR4
citation_journal_title=J. Glob. Optim.; citation_title=Convergence analysis of Taylor models and McCormick–Taylor models; citation_author=A Bompadre, A Mitsos, B Chachuat; citation_volume=57; citation_publication_date=2013; citation_pages=75-114; citation_doi=10.1007/s10898-012-9998-9; citation_id=CR5
citation_journal_title=Comput. Chem. Eng.; citation_title=A reduced space interior point strategy for optimization of differential algebraic systems; citation_author=AM Cervantes, A Wächter, RH Tütüncü, LT Biegler; citation_volume=24; citation_publication_date=2000; citation_pages=39-51; citation_doi=10.1016/S0098-1354(00)00302-1; citation_id=CR6
citation_journal_title=Ind. Eng. Chem. Res.; citation_title=Global methods for dynamic optimization and mixed-integer dynamic optimization; citation_author=B Chachuat, PI Barton, AB Singer; citation_volume=45; citation_issue=25; citation_publication_date=2006; citation_pages=8373-8392; citation_doi=10.1021/ie0601605; citation_id=CR7
Chachuat, B., Villanueva, M.: Bounding the solutions of parametric ODEs: when Taylor models meet differential inequalities. In: Bogle, I.D.L., Fairweather, M. (eds.) 22 European Symposium on Computer Aided Process Engineering, volume 30 of Comput. Aided Chem. Eng., pp. 1307–1311. Elsevier Science BV (2012)
Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. PhD thesis, University of Stockholm (1958)
citation_journal_title=J. Glob. Optim.; citation_title=The cluster problem in multivariate global optimization; citation_author=K Du, RB Kearfott; citation_volume=5; citation_issue=3; citation_publication_date=1994; citation_pages=253-265; citation_doi=10.1007/BF01096455; citation_id=CR10
citation_journal_title=Ind. Eng. Chem. Res.; citation_title=Global optimization for the parameter estimation of differential-algebraic systems; citation_author=WR Esposito, CA Floudas; citation_volume=39; citation_publication_date=2000; citation_pages=1291-1310; citation_doi=10.1021/ie990486w; citation_id=CR11
citation_title=Differential Equations with Discontinuous Righthand Sides; citation_publication_date=1988; citation_id=CR12; citation_author=AF Filippov; citation_publisher=Kluwer Academic Publishers
citation_title=Solving Ordinary Differential Equations I; citation_publication_date=1993; citation_id=CR13; citation_author=E Hairer; citation_author=SP Norsett; citation_author=G Wanner; citation_publisher=Springer
Harrison, G.: Dynamic models with uncertain parameters. In: Avula, X. (ed.) Proc. First Int. Conf. Math. Model., vol. 1, pp. 295–304. University of Missouri, Rolla (1977)
citation_journal_title=Math. Control Signal; citation_title=Efficient polyhedral enclosures for the reachable set of nonlinear control systems; citation_author=SM Harwood, PI Barton; citation_volume=28; citation_issue=1; citation_publication_date=2016; citation_pages=8; citation_doi=10.1007/s00498-015-0153-2; citation_id=CR15
citation_journal_title=IMA J. Math. Control I; citation_title=Bounds on reachable sets using ordinary differential equations with linear programs embedded; citation_author=SM Harwood, JK Scott, PI Barton; citation_volume=33; citation_issue=2; citation_publication_date=2016; citation_pages=519-541; citation_doi=10.1093/imamci/dnu054; citation_id=CR16
citation_journal_title=J. Process Control; citation_title=Robust optimization of nonlinear dynamic systems with application to a jacketed tubular reactor; citation_author=B Houska, F Logist, J Impe, M Diehl; citation_volume=22; citation_issue=6; citation_publication_date=2012; citation_pages=1152-1160; citation_doi=10.1016/j.jprocont.2012.03.008; citation_id=CR17
Houska, B., Villanueva, M.E., Chachuat, B.: A validated integration algorithm for nonlinear ODEs using Taylor models and ellipsoidal calculus. In: 52nd IEEE Conf. Decis. Control, pp. 484–489 (2013)
citation_journal_title=SIAM J. Numer. Anal.; citation_title=Stable set-valued integration of nonlinear dynamic systems using affine set-parameterizations; citation_author=B Houska, ME Villanueva, B Chachuat; citation_volume=53; citation_issue=5; citation_publication_date=2015; citation_pages=2307-2328; citation_doi=10.1137/140976807; citation_id=CR19
citation_title=Nonlinear Systems; citation_publication_date=2002; citation_id=CR20; citation_author=HK Khalil; citation_publisher=Prentice-Hall
citation_journal_title=J. Glob. Optim.; citation_title=Differentiable mccormick relaxations; citation_author=KA Khan, HAJ Watson, PI Barton; citation_volume=67; citation_issue=4; citation_publication_date=2017; citation_pages=687-729; citation_doi=10.1007/s10898-016-0440-6; citation_id=CR21
Krogh, B.H., Thorpe, C.E.: Integrated path planning and dynamic steering control for autonomous vehicles. In: Proceedings of 1986 IEEE Int. Conf. on Robot. Autom., vol. 3, pp. 1664–1669. IEEE (1986)
citation_journal_title=Comput. Chem. Eng.; citation_title=An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects; citation_author=DB Leineweber, I Bauer, HG Bock, JP Schlöder; citation_volume=27; citation_publication_date=2003; citation_pages=157-166; citation_doi=10.1016/S0098-1354(02)00158-8; citation_id=CR23
citation_journal_title=Ind. Eng. Chem. Res.; citation_title=Deterministic global optimization for parameter estimation of dynamic systems; citation_author=Y Lin, MA Stadtherr; citation_volume=45; citation_issue=25; citation_publication_date=2006; citation_pages=8438-8448; citation_doi=10.1021/ie0513907; citation_id=CR24
citation_journal_title=AIChE J.; citation_title=Deterministic global optimization of nonlinear dynamic systems; citation_author=Y Lin, MA Stadtherr; citation_volume=53; citation_issue=4; citation_publication_date=2007; citation_pages=866-875; citation_doi=10.1002/aic.11101; citation_id=CR25
citation_journal_title=Appl. Numer. Math.; citation_title=Validated solutions of initial value problems for parametric ODEs; citation_author=Y Lin, MA Stadtherr; citation_volume=57; citation_issue=10; citation_publication_date=2007; citation_pages=1145-1162; citation_doi=10.1016/j.apnum.2006.10.006; citation_id=CR26
citation_journal_title=Comput. Chem. Eng.; citation_title=Long term dynamic optimization of a catalytic reactor system; citation_author=I Løvik, M Hillestad, T Hertzberg; citation_volume=22; citation_publication_date=1998; citation_pages=S707-S710; citation_doi=10.1016/S0098-1354(98)00130-6; citation_id=CR27
citation_journal_title=Can. J. Chem. Eng.; citation_title=Multiplicity of solutions in the optimization of a bifunctional catalyst blend in a tubular reactor; citation_author=R Luus, J Dittrich, FJ Keil; citation_volume=70; citation_issue=4; citation_publication_date=1992; citation_pages=780-785; citation_doi=10.1002/cjce.5450700423; citation_id=CR28
citation_journal_title=Soft Comput.; citation_title=Multi-objective dynamic optimization with genetic algorithms for automatic parking; citation_author=D Maravall, J Lope; citation_volume=11; citation_issue=3; citation_publication_date=2007; citation_pages=249-257; citation_doi=10.1007/s00500-006-0066-6; citation_id=CR29
citation_journal_title=Math. Program.; citation_title=Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems; citation_author=GP McCormick; citation_volume=10; citation_issue=1; citation_publication_date=1976; citation_pages=147-175; citation_doi=10.1007/BF01580665; citation_id=CR30
citation_journal_title=Genome Res.; citation_title=Parameter estimation in biochemical pathways: a comparison of global optimization methods; citation_author=CG Moles, P Mendes, JR Banga; citation_volume=13; citation_issue=11; citation_publication_date=2003; citation_pages=2467-2474; citation_doi=10.1101/gr.1262503; citation_id=CR31
Moore, R.E.: Interval arithmetic and automatic error analysis in digital computing. PhD thesis, Stanford University (1962)
citation_title=Methods and Applications of Interval Analysis; citation_publication_date=1979; citation_id=CR33; citation_author=RE Moore; citation_publisher=SIAM
citation_title=Introduction to Interval Analysis; citation_publication_date=2009; citation_id=CR34; citation_author=RE Moore; citation_author=RB Kearfott; citation_author=MJ Cloud; citation_publisher=Society for Industrial and Applied Mathematics
citation_journal_title=Sitz.-Ber. Heidelberger Akad. Wiss. Math.-Naturwiss. Kl.; citation_title=Über die Eindeutigkeit der Integrale eines Systems gewöhnlicher Differentialgleichungen und die Konvergenz einer Gattung von Verfahren zur Approximation dieser Integrale; citation_author=M Müller; citation_volume=9; citation_publication_date=1927; citation_pages=3-38; citation_id=CR35
citation_journal_title=J. Glob. Optim.; citation_title=Convergence analysis of multivariate mccormick relaxations; citation_author=J Najman, A Mitsos; citation_volume=66; citation_issue=4; citation_publication_date=2016; citation_pages=597-628; citation_doi=10.1007/s10898-016-0408-6; citation_id=CR36
citation_journal_title=Reliab. Comput.; citation_title=Taylor forms-use and limits; citation_author=A Neumaier; citation_volume=9; citation_issue=1; citation_publication_date=2003; citation_pages=43-79; citation_doi=10.1023/A:1023061927787; citation_id=CR37
citation_journal_title=J. Aerosp. Comput. Inf. Commun.; citation_title=Real-time planning for multiple autonomous vehicles in dynamic uncertain environments; citation_author=A Pongpunwattana, R Rysdyk; citation_volume=1; citation_issue=12; citation_publication_date=2004; citation_pages=580-604; citation_doi=10.2514/1.12919; citation_id=CR38
citation_journal_title=Comput. Chem. Eng.; citation_title=Integrated scheduling and dynamic optimization of grade transitions for a continuous polymerization reactor; citation_author=A Prata, J Oldenburg, A Kroll, W Marquardt; citation_volume=32; citation_issue=3; citation_publication_date=2008; citation_pages=463-476; citation_doi=10.1016/j.compchemeng.2007.03.009; citation_id=CR39
citation_journal_title=BMC Bioinformatics; citation_title=Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems; citation_author=M Rodriguez-Fernandez, JA Egea, JR Banga; citation_volume=7; citation_publication_date=2006; citation_pages=483; citation_doi=10.1186/1471-2105-7-483; citation_id=CR40
Sahlodin, A.M.: Global optimization of dynamic process systems using complete search methods. PhD thesis, McMaster University (2013)
citation_journal_title=Comput. Chem. Eng.; citation_title=Convex/concave relaxations of parametric ODEs using Taylor models; citation_author=AM Sahlodin, B Chachuat; citation_volume=35; citation_issue=5; citation_publication_date=2011; citation_pages=844-857; citation_doi=10.1016/j.compchemeng.2011.01.031; citation_id=CR42
citation_journal_title=Appl. Numer. Math.; citation_title=Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs; citation_author=AM Sahlodin, B Chachuat; citation_volume=61; citation_issue=7; citation_publication_date=2011; citation_pages=803-820; citation_doi=10.1016/j.apnum.2011.01.009; citation_id=CR43
Schaber, S.D.: Tools for dynamic model development. PhD thesis, Massachusetts Institute of Technology (2014)
citation_journal_title=J. Glob. Optim.; citation_title=The theoretical and empirical rate of convergence for geometric branch-and-bound methods; citation_author=A Schöbel, D Scholz; citation_volume=48; citation_publication_date=2010; citation_pages=473-495; citation_doi=10.1007/s10898-009-9502-3; citation_id=CR45
citation_journal_title=J. Glob. Optim.; citation_title=Theoretical rate of convergence for interval inclusion functions; citation_author=D Scholz; citation_volume=53; citation_issue=4; citation_publication_date=2012; citation_pages=749-767; citation_doi=10.1007/s10898-011-9735-9; citation_id=CR46
Scott, J.K.: Reachability analysis and deterministic global optimization of differential-algebraic systems. PhD thesis, Massachusetts Institute of Technology (2012)
citation_journal_title=Comput. Chem. Eng.; citation_title=Tight, efficient bounds on the solutions of chemical kinetics models; citation_author=JK Scott, PI Barton; citation_volume=34; citation_issue=5; citation_publication_date=2010; citation_pages=717-731; citation_doi=10.1016/j.compchemeng.2009.11.021; citation_id=CR48
citation_journal_title=Automatica; citation_title=Bounds on the reachable sets of nonlinear control systems; citation_author=JK Scott, PI Barton; citation_volume=49; citation_issue=1; citation_publication_date=2013; citation_pages=93-100; citation_doi=10.1016/j.automatica.2012.09.020; citation_id=CR49
citation_journal_title=J. Optim. Theory Appl.; citation_title=Convex and concave relaxations for the parametric solutions of semi-explicit index-one differential-algebraic equations; citation_author=JK Scott, PI Barton; citation_volume=156; citation_publication_date=2013; citation_pages=617-649; citation_doi=10.1007/s10957-012-0149-8; citation_id=CR50
citation_journal_title=J. Glob. Optim.; citation_title=Improved relaxations for the parametric solutions of ODEs using differential inequalities; citation_author=JK Scott, PI Barton; citation_volume=57; citation_publication_date=2013; citation_pages=143-176; citation_doi=10.1007/s10898-012-9909-0; citation_id=CR51
citation_journal_title=Numer. Math.; citation_title=Interval bounds on the solutions of semi-explicit index-one DAEs. Part 1: analysis; citation_author=JK Scott, PI Barton; citation_volume=125; citation_issue=1; citation_publication_date=2013; citation_pages=1-25; citation_doi=10.1007/s00211-013-0531-y; citation_id=CR52
citation_journal_title=Numer. Math.; citation_title=Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation; citation_author=JK Scott, PI Barton; citation_volume=125; citation_issue=1; citation_publication_date=2013; citation_pages=27-60; citation_doi=10.1007/s00211-013-0532-x; citation_id=CR53
citation_journal_title=Optim. Control Appl. Methods; citation_title=Nonlinear convex and concave relaxations for the solutions of parametric ODEs; citation_author=JK Scott, B Chachuat, PI Barton; citation_volume=34; citation_issue=2; citation_publication_date=2013; citation_pages=145-163; citation_doi=10.1002/oca.2014; citation_id=CR54
citation_journal_title=J. Glob. Optim.; citation_title=Generalized McCormick relaxations; citation_author=JK Scott, MD Stuber, PI Barton; citation_volume=51; citation_issue=4; citation_publication_date=2011; citation_pages=569-606; citation_doi=10.1007/s10898-011-9664-7; citation_id=CR55
citation_journal_title=Comput. Chem. Eng.; citation_title=Rapid and accurate reachability analysis for nonlinear dynamic systems by exploiting model redundancy; citation_author=K Shen, JK Scott; citation_volume=106; citation_publication_date=2017; citation_pages=596-608; citation_doi=10.1016/j.compchemeng.2017.08.001; citation_id=CR56
citation_journal_title=SIAM J. Sci. Comput.; citation_title=Bounding the solutions of parameter dependent nonlinear ordinary differential equations; citation_author=AB Singer, PI Barton; citation_volume=27; citation_issue=6; citation_publication_date=2006; citation_pages=2167; citation_doi=10.1137/040604388; citation_id=CR57
citation_journal_title=J. Glob. Optim.; citation_title=Global optimization with nonlinear ordinary differential equations; citation_author=AB Singer, PI Barton; citation_volume=34; citation_issue=2; citation_publication_date=2006; citation_pages=159-190; citation_doi=10.1007/s10898-005-7074-4; citation_id=CR58
citation_journal_title=J. Phys. Chem. A; citation_title=Global dynamic optimization for parameter estimation in chemical kinetics; citation_author=AB Singer, JW Taylor, PI Barton, WH Green; citation_volume=110; citation_issue=3; citation_publication_date=2006; citation_pages=971-976; citation_doi=10.1021/jp0548873; citation_id=CR59
citation_journal_title=BIT Numer. Math.; citation_title=The logarithmic norm. History and modern theory; citation_author=G Söderlind; citation_volume=46; citation_issue=3; citation_publication_date=2006; citation_pages=631-652; citation_doi=10.1007/s10543-006-0069-9; citation_id=CR60
citation_journal_title=Ind. Eng. Chem. Res.; citation_title=Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equation systems; citation_author=I-B Tjoa, LT Biegler; citation_volume=30; citation_publication_date=1991; citation_pages=376-385; citation_doi=10.1021/ie00050a015; citation_id=CR61
citation_journal_title=Chem. Eng. Sci.; citation_title=Interval enclosures for reachable sets of chemical kinetic flow systems. Part 1: sparse transformation; citation_author=A Tulsyan, PI Barton; citation_volume=166; citation_publication_date=2017; citation_pages=334-344; citation_doi=10.1016/j.ces.2017.01.045; citation_id=CR62
citation_journal_title=Chem. Eng. Sci.; citation_title=Interval enclosures for reachable sets of chemical kinetic flow systems. Part 2: direct-bounding method; citation_author=A Tulsyan, PI Barton; citation_volume=166; citation_publication_date=2017; citation_pages=345-357; citation_doi=10.1016/j.ces.2016.12.021; citation_id=CR63
citation_journal_title=Chem. Eng. Sci.; citation_title=Interval enclosures for reachable sets of chemical kinetic flow systems. Part 3: indirect-bounding method; citation_author=A Tulsyan, PI Barton; citation_volume=166; citation_publication_date=2017; citation_pages=358-372; citation_doi=10.1016/j.ces.2017.02.047; citation_id=CR64
citation_journal_title=Ind. Eng. Chem. Res.; citation_title=Solution of a class of multistage dynamic optimization problems. 1. Problems without path constraints; citation_author=V Vassiliadis, R Sargent, C Pantelides; citation_volume=33; citation_publication_date=1994; citation_pages=2111-2122; citation_doi=10.1021/ie00033a014; citation_id=CR65
citation_journal_title=J. Glob. Optim.; citation_title=Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs; citation_author=ME Villanueva, B Houska, B Chachuat; citation_volume=62; citation_issue=3; citation_publication_date=2015; citation_pages=575-613; citation_doi=10.1007/s10898-014-0235-6; citation_id=CR66
citation_journal_title=J. Glob. Optim.; citation_title=The cluster problem revisited; citation_author=A Wechsung, SD Schaber, PI Barton; citation_volume=58; citation_issue=3; citation_publication_date=2014; citation_pages=429-438; citation_doi=10.1007/s10898-013-0059-9; citation_id=CR67