多目标优化问题近似解的非线性标量化刻画
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摘要
本文主要研究非线性标量化问题近似解与多目标优化问题近似解的关系.利用两种范数建立非线性标量化问题,得到了多目标优化问题近似有效解和近似真有效解的非线性标量化结果,并给出例子对主要结果进行了说明.
In this paper, we study the relationships between approximate solutions of scalar problems and approximate solutions of multiobjective programming. By utilizing two kind of norms to establish scalar problems, the nonlinear scalarizations of approximate efficient solutions and approximate properly efficient solutions for multiobjective programming problems are obtained. Moreover, some examples are given to explain the main results.
In this paper, we study the relationships between approximate solutions of scalar problems and approximate solutions of multiobjective programming. By utilizing two kind of norms to establish scalar problems, the nonlinear scalarizations of approximate efficient solutions and approximate properly efficient solutions for multiobjective programming problems are obtained. Moreover, some examples are given to explain the main results.
引文
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[3]White D J.Epsilon efficiency[J].J.Optim.The.Appl.,1986,49:319–337.
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[5]Gutierrez C,Jimenez B,Novo V.A unified approach and optimality conditions for approximate solutions of vector optimization problems[J].SIAM J.Optim.,2006,17(3):688–710.
[6]Gutierrez C,Jimenez B,Novo V.On approximate efficiency in multiobjective programming[J].Math.Meth.Oper.Res.,2006,64(1):165–185.
[7]Gao Y,Yang X M,Teo K L.Optimality conditions for approximate solutions of vector optimization problems[J].J.Indus.Mgt.Optim.,2011,7(2):483–496.
[8]Gao Y,Hou S H,Yang X M.Existence and optimality conditions for approximate solutions to vector optimization problems[J].J.Optim.The.Appl.,2012,152:97–120.
[9]Sawaragi Y,Nakayama H,Tanino T.Theory of multiobjective optimization[M].Japan:Dpt.Appl.Math.Konan Univ.,1985.
[10]Kaliszewski I.A theorem on nonconvex functions and its application to vector optimization[J].European J.Oper.Res.,1995,80:439–449.
[11]Beldiman M,Panaitescu E,Dogaru L.Approximate quasi efficient solutions in multiobjective optimization[J].Bull Math.Soc.Math.Roumanie Tome,2008,51(2):109–121.
[12]Rastegar N,Khorram E.A combined scalarizing method for multiobjective programming problems[J].European J.Oper.Res.,2014,236:229–237.
[13]李小燕,高英.多目标优化问题Proximal真有效解的最优性条件[J].应用数学和力学,2015,36(6):668–676.
[14]Zheng X Y.Scalarization of Henig proper efficient points in a normed space[J].J.Optim.The.Appl.,2000,105(1):233–247.
[15]杨丰梅,龚循华.C-单调范数与Henig有效点的切比雪夫标量化[J].系统科学与数学,2002,22(3):334–342.
[16]岳瑞雪,高英.多目标优化问题(ε,ε)-拟近似解的非线性标量化[J].数学杂志,2016,36(3):615–626.
[17]Gutierrez C,Huerga L,Novo V.Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems[J].J.Math.Anal.Appl.,2012,389:1046–1058.
[18]高英.多面体集下多目标优化问题近似解的若干性质[J].运筹学学报,2013,17(2):48–52.
[19]Flores-Baz′an F,Hern′andez E.A unified vector optimization problem:complete scalarizations and applications[J].Optim.,2011,60(12):1399–1419.
[20]郭辉.向量优化问题(C,ε)-弱有效解的一个非线性标量化性质[J].运筹学学报,2015,19(2):105–110.
[2]Loridan P.ε-solutions in vector minimization problems[J].J.Optim.The.Appl.,1984,43(2):265–276.
[3]White D J.Epsilon efficiency[J].J.Optim.The.Appl.,1986,49:319–337.
[4]Helbig S.One new concept forε-efficiency[R].Talk at Optimization Days,1992.
[5]Gutierrez C,Jimenez B,Novo V.A unified approach and optimality conditions for approximate solutions of vector optimization problems[J].SIAM J.Optim.,2006,17(3):688–710.
[6]Gutierrez C,Jimenez B,Novo V.On approximate efficiency in multiobjective programming[J].Math.Meth.Oper.Res.,2006,64(1):165–185.
[7]Gao Y,Yang X M,Teo K L.Optimality conditions for approximate solutions of vector optimization problems[J].J.Indus.Mgt.Optim.,2011,7(2):483–496.
[8]Gao Y,Hou S H,Yang X M.Existence and optimality conditions for approximate solutions to vector optimization problems[J].J.Optim.The.Appl.,2012,152:97–120.
[9]Sawaragi Y,Nakayama H,Tanino T.Theory of multiobjective optimization[M].Japan:Dpt.Appl.Math.Konan Univ.,1985.
[10]Kaliszewski I.A theorem on nonconvex functions and its application to vector optimization[J].European J.Oper.Res.,1995,80:439–449.
[11]Beldiman M,Panaitescu E,Dogaru L.Approximate quasi efficient solutions in multiobjective optimization[J].Bull Math.Soc.Math.Roumanie Tome,2008,51(2):109–121.
[12]Rastegar N,Khorram E.A combined scalarizing method for multiobjective programming problems[J].European J.Oper.Res.,2014,236:229–237.
[13]李小燕,高英.多目标优化问题Proximal真有效解的最优性条件[J].应用数学和力学,2015,36(6):668–676.
[14]Zheng X Y.Scalarization of Henig proper efficient points in a normed space[J].J.Optim.The.Appl.,2000,105(1):233–247.
[15]杨丰梅,龚循华.C-单调范数与Henig有效点的切比雪夫标量化[J].系统科学与数学,2002,22(3):334–342.
[16]岳瑞雪,高英.多目标优化问题(ε,ε)-拟近似解的非线性标量化[J].数学杂志,2016,36(3):615–626.
[17]Gutierrez C,Huerga L,Novo V.Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems[J].J.Math.Anal.Appl.,2012,389:1046–1058.
[18]高英.多面体集下多目标优化问题近似解的若干性质[J].运筹学学报,2013,17(2):48–52.
[19]Flores-Baz′an F,Hern′andez E.A unified vector optimization problem:complete scalarizations and applications[J].Optim.,2011,60(12):1399–1419.
[20]郭辉.向量优化问题(C,ε)-弱有效解的一个非线性标量化性质[J].运筹学学报,2015,19(2):105–110.