一种针对不确定性结构的区间鲁棒性优化方法
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摘要
本文提出一种针对不确定性结构的区间鲁棒性优化方法.首先,采用区间模型度量目标函数和约束函数中的不确定变量和参数.然后引入鲁棒性评价因子来度量目标函数的鲁棒性,并采用区间可能度方法(RPDI)处理不确定约束,进而建立区间鲁棒性优化模型.针对区间鲁棒性优化求解效率较低的问题,提出一种高效的优化方法,该方法将双层嵌套优化问题解耦为区间分析和确定性优化方法序列求解的问题.在每一迭代步,根据在当前设计点下的区间分析结果构建一个等效确定性优化问题,然后通过求解该问题更新设计点.此外,本文还提出一种迭代机制,来提高整个优化过程的收敛速度.最后,给出了三个数值算例和一个工程算例验证本文方法的有效性.
An interval robust optimization method for uncertain structures is proposed in this paper. First, interval model is adopted to quantify the uncertainties in variables and parameters of objective function and constraints. Second, a robust index is introduced to represent the robustness of objective function and reliability-based possibility degree of interval(RPDI) is employed to deal with uncertain constraints, based on which the interval robust optimization model is established. Third, for the low efficiency problem of solving the interval robust optimization, an efficient optimization method is proposed to decouple the double-loop nested optimization problem into a series of interval analyses and deterministic optimizations that sequentially solved. At each iteration, an equivalent deterministic optimization problem is constructed according to the interval analysis results at current design point and then the design point is updated by solving the constructed optimization problem. Besides, an iterative mechanism is proposed to improve the convergence rate of the whole optimization process. Finally, three numerical examples and an engineering application are utilized to demonstrate the accuracy and efficiency of the proposed method.
An interval robust optimization method for uncertain structures is proposed in this paper. First, interval model is adopted to quantify the uncertainties in variables and parameters of objective function and constraints. Second, a robust index is introduced to represent the robustness of objective function and reliability-based possibility degree of interval(RPDI) is employed to deal with uncertain constraints, based on which the interval robust optimization model is established. Third, for the low efficiency problem of solving the interval robust optimization, an efficient optimization method is proposed to decouple the double-loop nested optimization problem into a series of interval analyses and deterministic optimizations that sequentially solved. At each iteration, an equivalent deterministic optimization problem is constructed according to the interval analysis results at current design point and then the design point is updated by solving the constructed optimization problem. Besides, an iterative mechanism is proposed to improve the convergence rate of the whole optimization process. Finally, three numerical examples and an engineering application are utilized to demonstrate the accuracy and efficiency of the proposed method.
引文
1 Enevoldsen I,S?rensen J D.Reliability-based optimization in structural engineering.Struct Saf,1994,15:169-196
2 Kuschel N,Rackwitz R.Two basic problems in reliability-based structural optimization.Math Method Oper Res,1997,46:309-333
3 Kirjner-Neto C,Polak E,Kiureghian A D.An outer approximations approach to reliability-based optimal design of structures.J Optim Theory Appl,1998,98:1-16
4 Tu J,Choi K K,Park Y H.A new study on reliability-based design optimization.J Mech Des,1999,121:557-564
5 Du X,Chen W.Sequential optimization and reliability assessment method for efficient probabilistic design.J Mech Des,2004,126:225-233
6 Ben-Tal A,Nemirovski A.Robust optimization-methodology and applications.Math Program,2002,92:453-480
7 Bertsimas D,Pachamanova D,Sim M.Robust linear optimization under general norms.Oper Res Lett,2004,32:510-516
8 Mourelatos Z P,Liang J.A methodology for trading-off performance and robustness under uncertainty.J Mech Des,2005,128:856-863
9 Renaud J E.Automatic differentiation in robust optimization.AIAA J,1997,35:1072-1079
10 Taguchi G.Performance analysis design.Int J Prod Res,1978,16:521-530
11 Sandgren E,Cameron T M.Robust design optimization of structures through consideration of variation.Comput Struct,2002,80:1605-1613
12 Harzheim L,Warnecke U.Robustness optimization of the position of an anti-roll bar link to avoid the toggling of a rear axle stabilizer.Struct Multidisc Optim,2010,42:315-323
13 Liang J,Mourelatos Z P,Tu J.A single-loop method for reliability-based design optimisation.IJPD,2008,5:76-92
14 Mulvey J M,Vanderbei R J,Zenios S A.Robust optimization of large-scale systems.Oper Res,1995,43:264-281
15 El Ghaoui L,Lebret H.Robust solutions to least-squares problems with uncertain data.SIAM J Matrix Anal Appl,1997,18:1035-1064
16 Ben-Tal A,Nemirovski A,Roos C.Robust solutions of uncertain quadratic and conic-quadratic problems.SIAM J Optim,2002,13:535-560
17 Bertsimas D,Sim M.Robust discrete optimization and network flows.Math Program,2003,98:49-71
18 Xu H,Caramanis C,Mannor S.A distributional interpretation of robust optimization.Math OR,2012,37:95-110
19 Chen X,Fan J,Bian X.Structural robust optimization design based on convex model.Results Phys,2017,7:3068-3077
20 Chassein A,Goerigk M.Variable-sized uncertainty and inverse problems in robust optimization.Eur J Oper Res,2018,264:17-28
21 Beyer H G,Sendhoff B.Robust optimization-A comprehensive survey.Comput Method Appl Mech Eng,2007,196:3190-3218
22 Moore R E.Interval Analysis.Englewood Cliffs:Prentice-Hall,1966
23 Ben-Tal A,Nemirovski A.Robust solutions of uncertain linear programs.Oper Res Lett,1999,25:1-13
24 Sun W,Dong R,Xu H.A novel non-probabilistic approach using interval analysis for robust design optimization.J Mech Sci Technol,2009,23:3199-3208
25 Gunawan S,Azarm S.Non-gradient based parameter sensitivity estimation for single objective robust design optimization.J Mech Des,2003,126:395-402
26 Gunawan S,Azarm S.A feasibility robust optimization method using sensitivity region concept.J Mech Des,2004,127:858-865
27 Gunawan S,Azarm S.Multi-objective robust optimization using a sensitivity region concept.Struct Multidisc Optim,2005,29:50-60
28 Yonekura K,Kanno Y.Global optimization of robust truss topology via mixed integer semidefinite programming.Optim Eng,2010,11:355-379
29 Siddiqui S,Azarm S,Gabriel S.A modified benders decomposition method for efficient robust optimization under interval uncertainty.Struct Multidisc Optim,2011,44:259-275
30 Zhou J,Cheng S,Li M.Sequential quadratic programming for robust optimization with interval uncertainty.J Mech Des,2012,134:100913
31 Zhou J,Li M.Advanced robust optimization with interval uncertainty using a single-looped structure and sequential quadratic programming.JMech Des,2013,136:021008
32 Kang Z,Bai S.On robust design optimization of truss structures with bounded uncertainties.Struct Multidisc Optim,2013,47:699-714
33 Chen J,Tang Y.Sequential algorithms for structural design optimization under tolerance conditions.Eng Optimiz,2014,46:1183-1199
34 Jiang C,Xie H C,Zhang Z G,et al.A new interval optimization method considering tolerance design.Eng Optimiz,2015,47:1637-1650
35 Wang S,Qing X.A mixed interval arithmetic/affine arithmetic approach for robust design optimization with interval uncertainty.J Mech Des,2016,138:041403
36 Hu N,Duan B,Cao H,et al.Robust optimization with convex model considering bounded constraints on performance variation.Struct Multidisc Optim,2017,56:59-69
37 Cheng J,Liu Z,Tang M,et al.Robust optimization of uncertain structures based on normalized violation degree of interval constraint.Comput Struct,2017,182:41-54
38 Jiang C,Zhang Z G,Zhang Q F,et al.A new nonlinear interval programming method for uncertain problems with dependent interval variables.Eur J Oper Res,2014,238:245-253
39 Jiang C,Han X,Li D.A new interval comparison relation and application in interval number programming for uncertain problems.CMC-Comput Mater Con,2012,27:275-303
40 Qiu Z,Wang X.Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis.Int J Solids Struct,2005,42:4958-4970
41 Chen S H,Ma L,Meng G W,et al.An efficient method for evaluating the natural frequencies of structures with uncertain-but-bounded parameters.Comput Struct,2009,87:582-590
42 Qiu Z,Elishakoff I.Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis.Comput Method Appl Mech Eng,1998,152:361-372
43 Tang J C,Fu C M.A dimension-reduction interval analysis method for uncertain problems.CMES-Comp Model Eng Sci,2017,113:239-259
44 Gill P E,Murray W,Saunders M A.Snopt:An sqp algorithm for large-scale constrained optimization.SIAM Rev,2005,47:99-131
45 Marczyk J.Stochastic multidisciplinary improvement-beyond optimization.In:8th Symposium on Multidisciplinary Analysis and Optimization.Long Beach:AIAA,2000.2000-4929
46 Du X.Saddle point approximation for sequential optimization and reliability analysis.J Mech Design,2007,130:011011-11
47 Kurpati A,Azarm S,Wu J.Constraint handling improvements for multiobjective genetic algorithms.Struct Multidisc Optim,2002,23:204-213
48 Huang Z L,Zhou Y S,Jiang C,et al.Reliability-based multidisciplinary design optimization using incremental shifting vector strategy and its application in electronic product design.Acta Mech Sin,2018,34:285-302
2 Kuschel N,Rackwitz R.Two basic problems in reliability-based structural optimization.Math Method Oper Res,1997,46:309-333
3 Kirjner-Neto C,Polak E,Kiureghian A D.An outer approximations approach to reliability-based optimal design of structures.J Optim Theory Appl,1998,98:1-16
4 Tu J,Choi K K,Park Y H.A new study on reliability-based design optimization.J Mech Des,1999,121:557-564
5 Du X,Chen W.Sequential optimization and reliability assessment method for efficient probabilistic design.J Mech Des,2004,126:225-233
6 Ben-Tal A,Nemirovski A.Robust optimization-methodology and applications.Math Program,2002,92:453-480
7 Bertsimas D,Pachamanova D,Sim M.Robust linear optimization under general norms.Oper Res Lett,2004,32:510-516
8 Mourelatos Z P,Liang J.A methodology for trading-off performance and robustness under uncertainty.J Mech Des,2005,128:856-863
9 Renaud J E.Automatic differentiation in robust optimization.AIAA J,1997,35:1072-1079
10 Taguchi G.Performance analysis design.Int J Prod Res,1978,16:521-530
11 Sandgren E,Cameron T M.Robust design optimization of structures through consideration of variation.Comput Struct,2002,80:1605-1613
12 Harzheim L,Warnecke U.Robustness optimization of the position of an anti-roll bar link to avoid the toggling of a rear axle stabilizer.Struct Multidisc Optim,2010,42:315-323
13 Liang J,Mourelatos Z P,Tu J.A single-loop method for reliability-based design optimisation.IJPD,2008,5:76-92
14 Mulvey J M,Vanderbei R J,Zenios S A.Robust optimization of large-scale systems.Oper Res,1995,43:264-281
15 El Ghaoui L,Lebret H.Robust solutions to least-squares problems with uncertain data.SIAM J Matrix Anal Appl,1997,18:1035-1064
16 Ben-Tal A,Nemirovski A,Roos C.Robust solutions of uncertain quadratic and conic-quadratic problems.SIAM J Optim,2002,13:535-560
17 Bertsimas D,Sim M.Robust discrete optimization and network flows.Math Program,2003,98:49-71
18 Xu H,Caramanis C,Mannor S.A distributional interpretation of robust optimization.Math OR,2012,37:95-110
19 Chen X,Fan J,Bian X.Structural robust optimization design based on convex model.Results Phys,2017,7:3068-3077
20 Chassein A,Goerigk M.Variable-sized uncertainty and inverse problems in robust optimization.Eur J Oper Res,2018,264:17-28
21 Beyer H G,Sendhoff B.Robust optimization-A comprehensive survey.Comput Method Appl Mech Eng,2007,196:3190-3218
22 Moore R E.Interval Analysis.Englewood Cliffs:Prentice-Hall,1966
23 Ben-Tal A,Nemirovski A.Robust solutions of uncertain linear programs.Oper Res Lett,1999,25:1-13
24 Sun W,Dong R,Xu H.A novel non-probabilistic approach using interval analysis for robust design optimization.J Mech Sci Technol,2009,23:3199-3208
25 Gunawan S,Azarm S.Non-gradient based parameter sensitivity estimation for single objective robust design optimization.J Mech Des,2003,126:395-402
26 Gunawan S,Azarm S.A feasibility robust optimization method using sensitivity region concept.J Mech Des,2004,127:858-865
27 Gunawan S,Azarm S.Multi-objective robust optimization using a sensitivity region concept.Struct Multidisc Optim,2005,29:50-60
28 Yonekura K,Kanno Y.Global optimization of robust truss topology via mixed integer semidefinite programming.Optim Eng,2010,11:355-379
29 Siddiqui S,Azarm S,Gabriel S.A modified benders decomposition method for efficient robust optimization under interval uncertainty.Struct Multidisc Optim,2011,44:259-275
30 Zhou J,Cheng S,Li M.Sequential quadratic programming for robust optimization with interval uncertainty.J Mech Des,2012,134:100913
31 Zhou J,Li M.Advanced robust optimization with interval uncertainty using a single-looped structure and sequential quadratic programming.JMech Des,2013,136:021008
32 Kang Z,Bai S.On robust design optimization of truss structures with bounded uncertainties.Struct Multidisc Optim,2013,47:699-714
33 Chen J,Tang Y.Sequential algorithms for structural design optimization under tolerance conditions.Eng Optimiz,2014,46:1183-1199
34 Jiang C,Xie H C,Zhang Z G,et al.A new interval optimization method considering tolerance design.Eng Optimiz,2015,47:1637-1650
35 Wang S,Qing X.A mixed interval arithmetic/affine arithmetic approach for robust design optimization with interval uncertainty.J Mech Des,2016,138:041403
36 Hu N,Duan B,Cao H,et al.Robust optimization with convex model considering bounded constraints on performance variation.Struct Multidisc Optim,2017,56:59-69
37 Cheng J,Liu Z,Tang M,et al.Robust optimization of uncertain structures based on normalized violation degree of interval constraint.Comput Struct,2017,182:41-54
38 Jiang C,Zhang Z G,Zhang Q F,et al.A new nonlinear interval programming method for uncertain problems with dependent interval variables.Eur J Oper Res,2014,238:245-253
39 Jiang C,Han X,Li D.A new interval comparison relation and application in interval number programming for uncertain problems.CMC-Comput Mater Con,2012,27:275-303
40 Qiu Z,Wang X.Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis.Int J Solids Struct,2005,42:4958-4970
41 Chen S H,Ma L,Meng G W,et al.An efficient method for evaluating the natural frequencies of structures with uncertain-but-bounded parameters.Comput Struct,2009,87:582-590
42 Qiu Z,Elishakoff I.Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis.Comput Method Appl Mech Eng,1998,152:361-372
43 Tang J C,Fu C M.A dimension-reduction interval analysis method for uncertain problems.CMES-Comp Model Eng Sci,2017,113:239-259
44 Gill P E,Murray W,Saunders M A.Snopt:An sqp algorithm for large-scale constrained optimization.SIAM Rev,2005,47:99-131
45 Marczyk J.Stochastic multidisciplinary improvement-beyond optimization.In:8th Symposium on Multidisciplinary Analysis and Optimization.Long Beach:AIAA,2000.2000-4929
46 Du X.Saddle point approximation for sequential optimization and reliability analysis.J Mech Design,2007,130:011011-11
47 Kurpati A,Azarm S,Wu J.Constraint handling improvements for multiobjective genetic algorithms.Struct Multidisc Optim,2002,23:204-213
48 Huang Z L,Zhou Y S,Jiang C,et al.Reliability-based multidisciplinary design optimization using incremental shifting vector strategy and its application in electronic product design.Acta Mech Sin,2018,34:285-302