变序结构局部弱非控点的二阶刻画
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摘要
引进了一种二阶切导数,借助该切导数给出了变序结构集值优化问题取得局部弱非控点的二阶最优性必要条件.在某种特殊情况下,给出了一阶最优性条件.通过修正的Dubovitskij-Miljutin切锥导出的约束规格,给出了两个集值映射之和的二阶相依切导数的关系式,进一步得到目标函数与变锥函数的二阶相依切导数分开形式的最优性必要条件.
A kind of second-order tangent derivatives is introduced, with which a second-order necessary optimality condition is established for set-valued optimization with variable ordering structure in the sense of local weakly nondominated points. Under special circumstances, a first-order necessary optimality condition is obtained. The relationship to second-order contingent tangent derivatives for the sum of two set-valued maps is given under some constraint qualification indued by modified Dubovitskij-Miljutin tangent cones. Further more, a necessary optimality condition is obtained where the objective and constraining functions are considered separately with respect to second-order contingent tangent derivatives.
A kind of second-order tangent derivatives is introduced, with which a second-order necessary optimality condition is established for set-valued optimization with variable ordering structure in the sense of local weakly nondominated points. Under special circumstances, a first-order necessary optimality condition is obtained. The relationship to second-order contingent tangent derivatives for the sum of two set-valued maps is given under some constraint qualification indued by modified Dubovitskij-Miljutin tangent cones. Further more, a necessary optimality condition is obtained where the objective and constraining functions are considered separately with respect to second-order contingent tangent derivatives.
引文
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[6] Li S J, Zhu S K, Teo K L. Second-order Karush-Kuhn-Tucker optimality conditions for setvalued optimization[J]. Journal of Global Optimization, 2014, 58:673-692.
[7] Khanh P Q, Tung N M. Second-order conditions for open-cone minimizers and firm minimizers in set-valued optimization subject to mixed constraints[J]. Journal of Optimization Theory and Applications, 2016, 171:45-69.
[8] Nguyen Q H, Nguyen V T. New second-order optimality conditions for a class of differentiable optimization problems[J]. Journal of Optimization Theory and Applications, 2016, 171:27-44.
[9] Xu Y H, Li M, Peng Z H. A note on “Higher-order optimality in set-valued optimization using Studniarski derivatives and applications to duality”[Positivity. 18, 449-473(2014)][J].Positivity, 2016, 20:295-298.
[10] Peng Z H, Xu Y H. New second-order tangent epiderivatives and applications to set-valued optimization[J]. Journal of Optimization Theory and Applications, 2017, 172(1):128-140.
[11] Gutierrez C, Huerga L, Novo V, et al. Duality related to approximate proper solutions of vector optimization problems[J]. Journal of Global Optimization, 2016, 64:117-139.
[12] Aubin J P, Frankowska H. Set-Valued Analysis[M]. Birkhauser:Boston Massachusetts, 1990.
[13] Li S J, Zhu S K, Teo K L. New generalized second-order contingent epidervatives and set-valued optimization problems[J]. Journal of Optimization Theory and Applications, 2012, 152:587-604.
[14] Shi D S. Contingent derivatives of the perturbation map in multiobjective optimization[J].Journal of Optimization Theory and Applications, 1991, 70(2):385-396.
[2] Chen G Y, Yang X Q. Characterizations of variable domination structures via nonlinear scalarization[J]. Journal of Optimization Theory and Applications, 2002, 112(1):97-110.
[3] Durea M, Strugariu R, Tammer C. On set-valued optimization problems with variable ordering structure[J]. Journal of Global Optimization, 2015, 61:745-767.
[4] Jahn J, Khan A A, Zeilinger P. Second-order optimality conditions in set optimization[J].Journal of Optimization Theory and Applications, 2005, 125:331-347.
[5] Li S K, Teo K L, Yang X Q. Higher-order optimality conditions for set-valued optimization[J].Journal of Optimization Theory and Applications, 2008, 137:533-553.
[6] Li S J, Zhu S K, Teo K L. Second-order Karush-Kuhn-Tucker optimality conditions for setvalued optimization[J]. Journal of Global Optimization, 2014, 58:673-692.
[7] Khanh P Q, Tung N M. Second-order conditions for open-cone minimizers and firm minimizers in set-valued optimization subject to mixed constraints[J]. Journal of Optimization Theory and Applications, 2016, 171:45-69.
[8] Nguyen Q H, Nguyen V T. New second-order optimality conditions for a class of differentiable optimization problems[J]. Journal of Optimization Theory and Applications, 2016, 171:27-44.
[9] Xu Y H, Li M, Peng Z H. A note on “Higher-order optimality in set-valued optimization using Studniarski derivatives and applications to duality”[Positivity. 18, 449-473(2014)][J].Positivity, 2016, 20:295-298.
[10] Peng Z H, Xu Y H. New second-order tangent epiderivatives and applications to set-valued optimization[J]. Journal of Optimization Theory and Applications, 2017, 172(1):128-140.
[11] Gutierrez C, Huerga L, Novo V, et al. Duality related to approximate proper solutions of vector optimization problems[J]. Journal of Global Optimization, 2016, 64:117-139.
[12] Aubin J P, Frankowska H. Set-Valued Analysis[M]. Birkhauser:Boston Massachusetts, 1990.
[13] Li S J, Zhu S K, Teo K L. New generalized second-order contingent epidervatives and set-valued optimization problems[J]. Journal of Optimization Theory and Applications, 2012, 152:587-604.
[14] Shi D S. Contingent derivatives of the perturbation map in multiobjective optimization[J].Journal of Optimization Theory and Applications, 1991, 70(2):385-396.