向量集值优化问题的强有效性
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摘要
当目标函数和约束函数都是弧连通锥凸时,借助方向导数,利用择一性定理给出了约束向量优化问题取得强有效解的必要条件.利用强有效点的标量化定理给出了向量优化问题取得强有效解的Kuhn-Tucker最优性充分条件.引进了集值映射的强次梯度,并证明了强次梯度的存在性.作为应用,给出了带约束集值优化问题取得强有效元的充分条件和必要条件,得到了集值优化问题在强有效意义下的新刻画.
When both the objective function and constrained function are arc wise connected cone-convex functions, with directional derivative and alternative theorem, the necessary conditions are obtained for constrained vector-valued optimization problem to obtain its strongly efficient solutions. By using scalarization theorem for the strong efficient point, Kuhn-Tucker sufficient optimality condition is obtained for vector-valued optimization problem to obtain its strongly efficient solutions. And the strong-subgradient for set-valued map is introduced, whose existence theorem is proved. An equivalent depiction of the strong-subgradient is presented. As applications, the sufficient and necessary conditions for the set-valued optimization problem with constraint to attain its strongly efficient elements are given.
When both the objective function and constrained function are arc wise connected cone-convex functions, with directional derivative and alternative theorem, the necessary conditions are obtained for constrained vector-valued optimization problem to obtain its strongly efficient solutions. By using scalarization theorem for the strong efficient point, Kuhn-Tucker sufficient optimality condition is obtained for vector-valued optimization problem to obtain its strongly efficient solutions. And the strong-subgradient for set-valued map is introduced, whose existence theorem is proved. An equivalent depiction of the strong-subgradient is presented. As applications, the sufficient and necessary conditions for the set-valued optimization problem with constraint to attain its strongly efficient elements are given.
引文
[1]Li Z M. The optimization conditions for vector optimization set-valued maps[J]. Journal of Mathematical Analysis and Applications.,1999,237:413-424.
[2]Li Z F, Chen G Y. Lagrangian multipliers, saddle points and duality in vector optimization of set-vealued maps[J]. Journal of Mathematical Analysis and Applications,1997, 215:297-316.
[3]Li Z F, Wang S Y. Lagrange multipliers and saddle points in multiobjective programming [J]. Journal of Optimization Theory and Applications,1994,83:63-81.
[4]徐义红.集值优化问题的最优性条件[D].西安电子科技大学,2004.
[5]Xu Y H. Super efficiency in the nearly cone subconvexlike vector optimization with set-valued functions[J]. Acta.Math.Scientia,2005,46:193-211.
[6]Sach P H. Nearly subconvexlike set-valued maps and vector optimization problems [J]. Journal of Optimization Theory and Applications,2003,119(2):335-356.
[7]Borwein J M, Zhuang D.M.. Super efficiency in convex vector optimization [J]. Mathematics Methods of Operations Research,1991,35:105-122.
[8]Borwein J M, Zhuang D.. Super efficiency in vector optimization[J]. Transactions of the American mathematical society,1993,338(1):105-122.
[9]Zheng X Y. Proper efficiency in locally convex topological vector space[J]. Journal of Optimization Theory and Applications,1997,94:469-486.
[10]Zheng X Y. The dominaton property for efficiency in locally convex spaces[J]. Journal of Mathematical Analysis and Applications,1997,213:455-467.
[11]傅万涛,陈晓清.逼近锥族和严有效点[J].数学学报,1997,40(6)：933-938.
[12]傅万涛.赋范线性空间集合的严有效点[J].系统科学与数学,1997,17(4)：324-329.
[13]Cheng Y H, Fu W T. Strong efficiency in a locally convex space [J]. Mathematical Methods of Operations Research,1999,50(3):373-384.
[14]Li Z F. Benson proper efficiency in the vector optimization of set-valued maps [J]. Journal of Optimization Theory and Applications,1998,98:623-649.
[15]Song W. Lagrangian duality for minimization of nonconvex multifunctions[J]. Journal of Optimization Theory and Applications,1997,93:167-182.
[16]Yang X M, Yang X Q and Chen G Y. Theorem of the alternative and optimization with set-valued maps[J]. Journal of Optimization Theory and Applications,2000,107:627-640.
[17]Yang X M, Li D and Wang S Y. Near-Subconvexlikeness in vector optimization with set-valued functions [J]. Journal of Optimization Theory and Applications,2001, 110(2):413-427.
[18]Avriel M, Zang I. Generalized arcwise connected functions and characterization of local-global minimum properties [J]. Journal of Optimization Theory and Applications, 1980,32(4):407-425.
[19]Singh C. Elementary properties of arcwise connected sets and functions [J]. Journal of Optimization Theory and Applications,1983,41(2):377-387.
[20]Mukherjee R N, Yadav S R. A note on arcwise connected sets and functions [J]. Bulletin of the Australian Mathematical Society,1985,31:369-375.
[21]Bhatia D, Mehra A. Optimality conditions and duality involving arcwise connected and generalized arcwise connected functions [J]. Journal of Optimization Theory and Applications,1999,100(1):181-194.
[22]Fu J Y, Wang Y H. Arcwise connected cone-convex functions and mathematical programming [J]. Journal of Optimization Theory and Applications,2003,118(2):339-352.
[23]盛宝怀,刘三阳.用广义梯度刻画集值优化Benson真有效解[J].应用数学学报,2002,25(1)：22-28.
[24]李声杰.S-凸集值映射的次梯度和弱有效解[J].高校应用数学学报,1998,13(4)：463-472.
[25]盛宝怀,刘三阳.Benson真有效意义下集值优化的广义最优性条件[J].数学学报,2003,46(3)：611-620.
[26]Li T Y, Xu Y H. The strictly efficient subgradient of set-valued optimization[J]. Bull. Austral. Math. Soc.,2007,75:361-371.
[27]Jahn J. Mathematical vector optimization in partially-ordered linear spaces [M]. Verlag Peter Lang, Frankfurt am Main, Germany,1986.
[28]Hayashi M, Komiya H. Perfect duality for convexlike programs [J]. Journal of Optimization Theory and Applications,1982,38(2):179-189.
[29]Jeyakumar V. Convexlike alternative theorems and mathematical programming [J]. Optimization,1985,16:643-652.
[30]武育楠,戎卫东.集值映射向量优化问题的强有效性[J].内蒙古大学学报(自然科学版),1999,30(4)：415-421.
[31]林锉云,董加礼.多目标规划的方法与理论[M].长春：吉林教育出版社,1992.
[32]胡毓达.多目标规划有效性理论[M].上海：上海科学技术出版社,1994.
[33]赵春英,戎卫东.ε-强(严)有效点集的标量化[C].中国运筹学会第八届学术交流会论文集.深圳：2006,220-225.
[34]Hu Y D, Ling C. Connectedness of cone super efficient point sets in locally convex topological spaces [J]. Journal of Optimization Theory and Applications,2000,107(2): 433-446.
[35]徐义红.近似锥-次类凸集值优化问题的强有效性[J].南昌大学学报：理科版,2003,27(2)：121-124.
[36]Rockafeller R T. Extension of subgradient calculas with applications to optimization[J]. Nonlinear Analysis Theory and Applications.1985,9(7):665-698.
[37]Clarke F. H. Optimization and Nonsmooth Analysis[M]. New York:Wiley-Inter science, 1983.
[38]Papagergiou N S. Nonsmooth analysis on partially ordered vector space:Part 2-Nonconvex case, Clarke's Theorem[J]. Pacific J. Math 1983,109(2):463-495.
[39]侯震梅,周勇.集值映射的广义梯度与超有效解[J].吉林大学学报(理学版),2006,44(1)：9-14.
[40]Li Z. A theorem of the alternative and its application to the optimization of set-valued maps[J]. J. Optim. Theory Appl.,1999,100(2):365-375.
[41]Gong X. H. Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior[J]. J. Math. Anal. Appl.2005,307,12-31.
[42]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.
[43]Qiu J H. Superefficiency in local convex spaces[J]. J. Optim. Theory Appl.,2007,135:19-35.
[44]侯震梅.集值优化最优性条件与稳定性问题的研究[D].西安电子科技大学,2005.
[45]徐义红.集值优化问题强有效解的Kuhn-Tucker最优性条件[J].数学研究与评论,2006,26(2)354-360.
[46]李太勇,徐义红.不可微向量优化问题严有效解的最优性条件[J].运筹学学报,2008,12(1)：1-8.
[47]李太勇,徐义红.内部锥次类图集值优化问题严有效解的最优性条件[J].南昌大学学报(理科版),2007,31(4)：327-331.
[48]程永红. 局部凸空间中的强有效性[D].南昌大学,1997.
[49]Henig M I. Proper efficiency with respect to cones[J]. J. Optim. Theory Appl.,1982,36 (3):387-407.
[50]Sach P H. New generalized convexity notion for set-valued maps and application to vector optimization [J]. Journal of Optimization Theory and Applications,2005,125(1)157-179.
[2]Li Z F, Chen G Y. Lagrangian multipliers, saddle points and duality in vector optimization of set-vealued maps[J]. Journal of Mathematical Analysis and Applications,1997, 215:297-316.
[3]Li Z F, Wang S Y. Lagrange multipliers and saddle points in multiobjective programming [J]. Journal of Optimization Theory and Applications,1994,83:63-81.
[4]徐义红.集值优化问题的最优性条件[D].西安电子科技大学,2004.
[5]Xu Y H. Super efficiency in the nearly cone subconvexlike vector optimization with set-valued functions[J]. Acta.Math.Scientia,2005,46:193-211.
[6]Sach P H. Nearly subconvexlike set-valued maps and vector optimization problems [J]. Journal of Optimization Theory and Applications,2003,119(2):335-356.
[7]Borwein J M, Zhuang D.M.. Super efficiency in convex vector optimization [J]. Mathematics Methods of Operations Research,1991,35:105-122.
[8]Borwein J M, Zhuang D.. Super efficiency in vector optimization[J]. Transactions of the American mathematical society,1993,338(1):105-122.
[9]Zheng X Y. Proper efficiency in locally convex topological vector space[J]. Journal of Optimization Theory and Applications,1997,94:469-486.
[10]Zheng X Y. The dominaton property for efficiency in locally convex spaces[J]. Journal of Mathematical Analysis and Applications,1997,213:455-467.
[11]傅万涛,陈晓清.逼近锥族和严有效点[J].数学学报,1997,40(6)：933-938.
[12]傅万涛.赋范线性空间集合的严有效点[J].系统科学与数学,1997,17(4)：324-329.
[13]Cheng Y H, Fu W T. Strong efficiency in a locally convex space [J]. Mathematical Methods of Operations Research,1999,50(3):373-384.
[14]Li Z F. Benson proper efficiency in the vector optimization of set-valued maps [J]. Journal of Optimization Theory and Applications,1998,98:623-649.
[15]Song W. Lagrangian duality for minimization of nonconvex multifunctions[J]. Journal of Optimization Theory and Applications,1997,93:167-182.
[16]Yang X M, Yang X Q and Chen G Y. Theorem of the alternative and optimization with set-valued maps[J]. Journal of Optimization Theory and Applications,2000,107:627-640.
[17]Yang X M, Li D and Wang S Y. Near-Subconvexlikeness in vector optimization with set-valued functions [J]. Journal of Optimization Theory and Applications,2001, 110(2):413-427.
[18]Avriel M, Zang I. Generalized arcwise connected functions and characterization of local-global minimum properties [J]. Journal of Optimization Theory and Applications, 1980,32(4):407-425.
[19]Singh C. Elementary properties of arcwise connected sets and functions [J]. Journal of Optimization Theory and Applications,1983,41(2):377-387.
[20]Mukherjee R N, Yadav S R. A note on arcwise connected sets and functions [J]. Bulletin of the Australian Mathematical Society,1985,31:369-375.
[21]Bhatia D, Mehra A. Optimality conditions and duality involving arcwise connected and generalized arcwise connected functions [J]. Journal of Optimization Theory and Applications,1999,100(1):181-194.
[22]Fu J Y, Wang Y H. Arcwise connected cone-convex functions and mathematical programming [J]. Journal of Optimization Theory and Applications,2003,118(2):339-352.
[23]盛宝怀,刘三阳.用广义梯度刻画集值优化Benson真有效解[J].应用数学学报,2002,25(1)：22-28.
[24]李声杰.S-凸集值映射的次梯度和弱有效解[J].高校应用数学学报,1998,13(4)：463-472.
[25]盛宝怀,刘三阳.Benson真有效意义下集值优化的广义最优性条件[J].数学学报,2003,46(3)：611-620.
[26]Li T Y, Xu Y H. The strictly efficient subgradient of set-valued optimization[J]. Bull. Austral. Math. Soc.,2007,75:361-371.
[27]Jahn J. Mathematical vector optimization in partially-ordered linear spaces [M]. Verlag Peter Lang, Frankfurt am Main, Germany,1986.
[28]Hayashi M, Komiya H. Perfect duality for convexlike programs [J]. Journal of Optimization Theory and Applications,1982,38(2):179-189.
[29]Jeyakumar V. Convexlike alternative theorems and mathematical programming [J]. Optimization,1985,16:643-652.
[30]武育楠,戎卫东.集值映射向量优化问题的强有效性[J].内蒙古大学学报(自然科学版),1999,30(4)：415-421.
[31]林锉云,董加礼.多目标规划的方法与理论[M].长春：吉林教育出版社,1992.
[32]胡毓达.多目标规划有效性理论[M].上海：上海科学技术出版社,1994.
[33]赵春英,戎卫东.ε-强(严)有效点集的标量化[C].中国运筹学会第八届学术交流会论文集.深圳：2006,220-225.
[34]Hu Y D, Ling C. Connectedness of cone super efficient point sets in locally convex topological spaces [J]. Journal of Optimization Theory and Applications,2000,107(2): 433-446.
[35]徐义红.近似锥-次类凸集值优化问题的强有效性[J].南昌大学学报：理科版,2003,27(2)：121-124.
[36]Rockafeller R T. Extension of subgradient calculas with applications to optimization[J]. Nonlinear Analysis Theory and Applications.1985,9(7):665-698.
[37]Clarke F. H. Optimization and Nonsmooth Analysis[M]. New York:Wiley-Inter science, 1983.
[38]Papagergiou N S. Nonsmooth analysis on partially ordered vector space:Part 2-Nonconvex case, Clarke's Theorem[J]. Pacific J. Math 1983,109(2):463-495.
[39]侯震梅,周勇.集值映射的广义梯度与超有效解[J].吉林大学学报(理学版),2006,44(1)：9-14.
[40]Li Z. A theorem of the alternative and its application to the optimization of set-valued maps[J]. J. Optim. Theory Appl.,1999,100(2):365-375.
[41]Gong X. H. Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior[J]. J. Math. Anal. Appl.2005,307,12-31.
[42]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.
[43]Qiu J H. Superefficiency in local convex spaces[J]. J. Optim. Theory Appl.,2007,135:19-35.
[44]侯震梅.集值优化最优性条件与稳定性问题的研究[D].西安电子科技大学,2005.
[45]徐义红.集值优化问题强有效解的Kuhn-Tucker最优性条件[J].数学研究与评论,2006,26(2)354-360.
[46]李太勇,徐义红.不可微向量优化问题严有效解的最优性条件[J].运筹学学报,2008,12(1)：1-8.
[47]李太勇,徐义红.内部锥次类图集值优化问题严有效解的最优性条件[J].南昌大学学报(理科版),2007,31(4)：327-331.
[48]程永红. 局部凸空间中的强有效性[D].南昌大学,1997.
[49]Henig M I. Proper efficiency with respect to cones[J]. J. Optim. Theory Appl.,1982,36 (3):387-407.
[50]Sach P H. New generalized convexity notion for set-valued maps and application to vector optimization [J]. Journal of Optimization Theory and Applications,2005,125(1)157-179.