向量多项式优化问题的数值方法
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摘要
向量多项式优化问题中的目标函数和约束条件都是由多项式描述的.先将多目标多项式函数分别通过主要目标法、线性加权和法和理想点法等转化为单目标多项式函数,再利用Lasserre松弛方法求解该多项式优化问题,从而得到原向量多项式优化问题的弱有效解或有效解.数值实验结果表明该数值方法是有效的.
For vector polynomial optimization problem, the objective function and constraint condition are all described in polynomial. Firstly, the multi-objective polynomial function is transformed into the single-objective polynomial function by using the main objective method, linear weighted sum method and ideal point method respectively. Then Lasserre relaxation method is applied to solve the polynomial optimization problem, and its optimal solution is obtained, which is also the weak efficient solution or effective solution of the original vector polynomial optimization problem. Numerical experiments show that the proposed numerical methods are effective.
For vector polynomial optimization problem, the objective function and constraint condition are all described in polynomial. Firstly, the multi-objective polynomial function is transformed into the single-objective polynomial function by using the main objective method, linear weighted sum method and ideal point method respectively. Then Lasserre relaxation method is applied to solve the polynomial optimization problem, and its optimal solution is obtained, which is also the weak efficient solution or effective solution of the original vector polynomial optimization problem. Numerical experiments show that the proposed numerical methods are effective.
引文
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[16] HENDRIX ELIGIUS M T,G-TóTH BOGLáRKA.Introduction to Nonlinear and Global Optimization[M].New York:Springer,2010:137-170.
[17] HENRION DIDIER,LASSERRE JEAN-BERBARD,L?FBERG JOHAN.GloptiPoly 3:Moments,Optimization and Semidefinite Programming[J].Optimization Methods and Software,2009,24(4-5):761-779.
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[19] FLOUDAS CHRISTODOULOS A,PARDALOS PANOS M.A Collection of Test Problems for Constrained Global Optimization Algorithms[M]//Lecture Notes in Computer Science.Berlin:Springer-Verlag,1990:1-35.
[20] NIE Jiawang.Linear Optimization with Cones of Moments and Nonnegative Polynomials[J].Mathematical Programming,2015,153(1):247-274.
[21] NIE Jiawang,WANG Li,YE JANE J.Bilevel Polynomial Programs and Semidefinite Relaxation Methods[J].SIAM Journal on Optimization,2017,27(3):1 728-1 757.
[22] WEISSER TILLMANN,LASSERRE JEAN BERNARD,TOH KIM-CHUAN.Sparse-BSOS:A Bounded Degree SOS Hierarchy for Large Scale Polynomial Optimization with Sparsity[J].Mathematical Programming Computation,2018,10(1):1-32.
[2] PARETO VILFREDO.Manuale di Economia Politica[M].Milano:Societ a Editrice Linraria,1906:1-27.
[3] KOOPMANS TJALLING C.Analysis of Production as an Efficient Combination of Activities[J].Analysis of Production and Allocation,1951,158(1):33-97.
[4] KUHN HAROLD W,TUCKER ALBERT WILLIAM.Nonliear Programming[C].Proceeding of the Second Berkeley Symposium on Mathematical Statisitics and Probability.California:University of California Press,1953,35(2):481-492.
[5] 林锉云,董加礼.多目标优化的方法与理论[M].吉林:吉林教育出版社,1992:55-72.
[6] GIANNESSI FRANCO,MASTROENI GIANDOMENICO,PELLEGRINI LETIZIA.On the Theory of Vector Optimization and Variational Inequalities.Image Space Analysis and Separation[J].Vector Variational Inequalities and Vector Equilibria:Mathematical Theories,2000,38:153-215.
[7] 陈光亚.向量优化问题某些基础理论及其发展[J].重庆师范大学学报(自然科学版),2005,22(3):6-9.
[8] JAHN JOHANNES.Vector Optimization:Theory,Applications and Extensions[M].New York:SpringerVerlag,2011:190-207.
[9] FLORES-BAZáN FABIáN,HERNáNDEZ ELVIRA.A Unified Vector Optimization Problem:Complete Scalarizations and Applications[J].Optimization,2011,60(12):1 399-1 419.
[10] GUTIéRREZ CéSAR,HUERGA L,JIMéNEZ B,et al.Approximate Solutions of Vector Optimization Problems via Improvement Sets in Real Linear Spaces[J].Journal of Global Optimization,2018,70(4):875-901.
[11] FLOUDAS CHRISTODOULOS A,PARDALOS PANOS M.Recent Advances in Global Optimization[M].Princeton University Press,1992:165-199.
[12] HORST REINER,PARDALOS PANOS M.Handbook of Global Optimization[J].Nonconvex Optimization and Its Applications,2002,2(5):217-269.
[13] RUBINOV ALEXANDER.Abstract Convexity and Global Optimization[M].Berlin:Kluwer Academic Publishers,2000:450-470.
[14] LASSERRE JEAN BERNARD.Global Optimization with Polynomials and the Problem of Moments[J].SIAM J Optim,2001,11(3):796-817.
[15] FLOUDAS CHRISTODOULOS A,PARDALOS P?NOS M,ADJIMAN CLAIRE,et al.Handbook of Test Problems in Local and Global Optimization[M]//Nonconvex Optimization and Its Applications.Berlin:Kluwer Academic Publishers,1999:1-31.
[16] HENDRIX ELIGIUS M T,G-TóTH BOGLáRKA.Introduction to Nonlinear and Global Optimization[M].New York:Springer,2010:137-170.
[17] HENRION DIDIER,LASSERRE JEAN-BERBARD,L?FBERG JOHAN.GloptiPoly 3:Moments,Optimization and Semidefinite Programming[J].Optimization Methods and Software,2009,24(4-5):761-779.
[18] LASSERRE JEAN BERNARD.Moments,Positive Polynomials and Their Applications[M].Imperial College Press,2010:1-144.
[19] FLOUDAS CHRISTODOULOS A,PARDALOS PANOS M.A Collection of Test Problems for Constrained Global Optimization Algorithms[M]//Lecture Notes in Computer Science.Berlin:Springer-Verlag,1990:1-35.
[20] NIE Jiawang.Linear Optimization with Cones of Moments and Nonnegative Polynomials[J].Mathematical Programming,2015,153(1):247-274.
[21] NIE Jiawang,WANG Li,YE JANE J.Bilevel Polynomial Programs and Semidefinite Relaxation Methods[J].SIAM Journal on Optimization,2017,27(3):1 728-1 757.
[22] WEISSER TILLMANN,LASSERRE JEAN BERNARD,TOH KIM-CHUAN.Sparse-BSOS:A Bounded Degree SOS Hierarchy for Large Scale Polynomial Optimization with Sparsity[J].Mathematical Programming Computation,2018,10(1):1-32.