摘要
向量优化理论与方法作为最优化理论及应用研究的一个重要方向,近年来发展迅速,已成为国际优化领域研究的热点之一.这一问题的研究涉及凸分析、非光滑分析、非线性分析等多门分支学科.同时,它在经济管理、工程设计、交通运输、生态保护以及最优决策等诸多领域都具有广泛应用.本文主要致力于向量优化问题解的性质的四个方面的研究：集值向量优化问题解的统一性及相关性质研究；非光滑意义下C(T)值向量优化问题的最优性条件研究；非光滑意义下向量优化问题的正则性条件与(弱)强Kuhn-Tucker最优性必要条件研究以及广义凸性条件下非线性优化问题解集的刻画研究等.本文共分为五章,主要内容如下：
1.第一章简要叙述了向量优化理论与应用研究的背景和意义,对向量优化理论及与本文相关的研究方向的发展历史与研究现状进行了综述.介绍了本文相关研究工作所需要的一些基本概念和基础理论,进而提出了本文所要研究的主要内容.
2.第二章研究了集值向量优化问题解的统一性及其相关性质.首先,我们给出改进集的一些重要性质并利用改进集提出了一类新的广义凸性函数—邻近E-次似凸函数,并建立了相应的择一性定理；其次,基于Benson真有效性和近似Benson真有效性的思想,利用改进集提出了一类统一的真有效性概念-E-Benson真有效性,并在邻近E-次似凸性假设下建立了集值向量优化问题E-Benson真有效性的标量化定理、拉格朗日乘子定理、鞍点定理以及弱对偶定理与强对偶定理；然后,基于弱有效性与近似弱有效性的思想,利用改进集提出了一类统一的弱有效性概念—E-弱有效性,并在邻近E-次似凸性假设下建立了集值向量优化问题E-弱有效性的标量化定理、拉格朗日乘子定理、鞍点定理以及弱对偶定理与强对偶定理；最后,我们利用改进集E获得了向量优化问题的一个统一的稳定性结果.
3.第三章利用Clarke方向导数和Clarke次微分研究了C(T)值向量优化问题的一些最优性条件.首先,我们在伪凸性假设下利用Clarke方向导数和Clarke次微分给出了G(T)值向量优化问题有效解的一个充分条件和弱有效解的一个等价条件；其次,我们推广了相应结果到带不等式约束的C(T)值向量优化问题的情形；最后,我们在有限维空间中讨论了有效解的充分性条件的一个特例并利用线性化锥给出了该充分性条件的等价形式.
4.第四章研究了正则性条件与向量优化问题的Kuhn-Tucker最优性必要条件.首先,我们利用Clarke方向导数提出了新的正则性条件,并研究了与其它正则性条件之间的一些关系；然后,在新的正则性条件下获得了非光滑向量优化问题有效解的弱Kuhn-Tucker最优性必要条件与Geoffrion真有效解的强Kuhn-Tucker最优性必要条件；最后,我们在η-伪线性假设条件下给出了可微向量优化问题的有效解的一些充分必要条件.
5.第五章研究了广义凸性条件下一些非光滑优化问题最优解集的刻画.首先,我们给出了局部Lipschitz77-伪线性函数的一些性质,并研究了非光滑77-伪线性优化问题最优解集的刻画；其次,我们指出了国际上关于非光滑伪不变凸优化问题最优解集刻画研究中的一些不足之处,并修正了相应结果；最后,我们利用拉格朗日乘子方法研究了带不等式约束的非光滑伪凸优化和非光滑伪不变凸优化问题最优解集的刻画.
In recent years, as an important research scope of optimization theory and applica-tions, vector optimization theory and methods have been developed rapidly and become one of the main research fields in optimization. Study on which involves many disci-plines, such as: convex analysis, nonsmooth analysis, nonlinear analysis, and so on. At the same time, it has been playing an important role in many fields, such as:economics and management, engineering design, transportation, environmental protection and op-timal design, etc. We mainly focus on characterizations of solutions of vector optimiza-tion problems in four aspects:unified solution concepts and some characterizations of solutions of vector optimization problems with set-valued maps, optimality conditions of nonsmooth vector optimization problems with C(T)-valued maps, regularity conditions and (weak) strong Kuhn-Tucker optimality necessary conditions for nonsmooth vector optimization problems and characterizations of solution sets for nonlinear optimization problems with generalized convexity. This thesis includes five chapters as follows:
1. In chapter1, we first give some brief introductions to the background and signif-icance of vector optimization theory and applications, especially for studying on characterizations of solutions. And we also summarize the developments of study-ing on vector optimization theory and methods in five aspects associated with this thesis. Secondly, we recall some basic concepts and results which will be used in this thesis. Finally, we outline the contents studied in this thesis.
2. In chapter2, we are devoted to study some unified solution concepts and some related characterizations of solutions of vector optimization problems with set-valued maps. We first give some important characterizations of improvement set. And then, we propose a class of new generalized convexity named as nearly E-subconvexlikeness via improvement sets and establish an alternative theorem under the nearly-subconvexlikeness. Secondly, based on the ideas of Benson proper efficiency and approximate Benson proper efficiency, we propose a class of unified proper efficiency named as E-Benson proper efficiency via improvement sets and with the assumption of nearly.E-subconvexlikeness, establish scalarization theo-rem, Lagrange multipliers theorem, saddle points theorem, weak duality theorem and strong duality theorem of jE-Benson proper efficiency. Thirdly, based on the ideas of weak efficiency and approximate weak efficiency, we propose a class of unified weak efficiency named as weak E-efficiency via improvement sets and with the as-sumption of nearly E-subconvexlikeness, establish scalarization theorem, Lagrange multipliers theorem, saddle points theorem, weak duality theorem and strong dual-ity theorem of weak E-efficiency. At last, we also obtain a unified stability result in terms of improvement sets for vector optimization problems.
3. In chapter3, we study some optimality conditions for vector optimization problems with C(T)-valued maps by Clarke directional derivatives and Clarke subdifferentials. Firstly, under the pseudoconvexity and by Clarke directional derivatives and Clarke subdifferentials, we give a sufficient condition of efficient solution and an equivalent condition of weak efficient solution for vector optimization problems with C(T)-valued maps. Secondly, we generalize the corresponding results to the case of vector optimization problems with C(T)-valued maps and inequality constraints. At last, in finite space, we discuss a special case of sufficient condition of efficient solution and give an equivalent version of the sufficient condition by using linear cone.
4. In chapter4, we study the Kuhn-Tucker necessary optimality conditions of vector optimization problems under the regularity conditions. Firstly, we propose new reg-ularity conditions by Clarke directional derivatives and discuss some relations with some other known regularity conditions. Under the new regularity condition, we obtain weak Kuhn-Tucker necessary optimality condition of efficient solution and strong Kuhn-Tucker necessary optimality condition of Geoffrion proper efficient so-lution for nonsmooth vector optimization problems. Moreover, under the assump-tion of η-pseudolinearity, we also give some necessary and sufficient conditions of efficient solution for differentiable vector optimization problems.
5. In chapter5, we study characterizations of the optimal solution sets for some classes of nonsmooth optimization problems under the suitable generalized con-vexity. Firstly, under the η-pseudolinearity, we give some properties of locally Lip-schitz77-pseudolinear function and characterize the optimal solution sets of locally Lipschitz η-pseudolinear optimization problems. Secondly, we point out some dis-advantages of some known research results on characterizations of the solution sets of nonsmooth pseudoinvex optimization problems and revise them accordingly. In the end, by Lagrange multipliers method, we also give some characterizations of the optimal solution sets for nonsmooth pseudoconvex optimization problems and nonsmooth pseudoinvex optimization problems with inequality constraints.
引文
[1]E. Johnsen. Studies in multiobjective decision models[M]. Lund Sweden:Economic Research Center in Lund,1968.
[2]D.T. Luc. Theory of vector optimization[M]. Lecture Notes in Economics and Mathematical Sciences,319, Springer-Verlag, Berlin,1989.
[3]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.
[4]A. Gopfert, C. Tammer, H. Riahi and C. Zalinescu. Variational methods in partially ordered spaces[M]. Springer-Verlag, New York,2003.
[5]J. Jahn. Vector optimization:theory, applications, and extensions[M]. Springer-Verlag, Berlin,2004.
[6]G.Y. Chen, X.X. Huang and X.Q. Yang. Vector optimization[M]. Lecture Notes in Economics and Mathematical Sciences,541, Springer-Verlag, Berlin,2005.
[7]M. Ehrgott. Multicriteria optimization[M]. Springer-Verlag, Berlin,2005.
[8]S.K. Mishra, S.Y. Wang and K.K. Lai. Generalized convexity and vector optimiza-tion[M]. In:Nonconvex Optimization and Its Applications,90, Springer-Verlag, Berlin,2009.
[9]V. Pareto. Manuale di economic political[M]. Malano:Societa Editrice,1906.
[10]T.C. Koopmans. Analysis of production as an efficient combination of activities[C]. In:T.C. Koopmans, Activity Analysis of Production and Allocation. Cowles Com-mission Monograph. New York:John Wiley and Sons,1951,13:33-97.
[11]H.W. Kuhn and A.W. Tucker. Nonlinear programming[C]. In:Proceeding of the second berley symposium on mathematical statisties and probability. California: University of California press,1951:481-492.
[12]A.M. Geoffrion. Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications [J],1968,22:618-630.
[13]J. Borwein. Proper efficient points for maximizations with respect to cones. SIAM Journal on Control and Optimization [J],1977,15(1):57-63.
[14]R. Hartley. On cone-efficiency, cone-convexity and cone-compactness. SIAM Jour-nal of Applied Mathematics [J],1978,34(2):211-222.
[15]H.P. Benson. An improved definition of proper efficiency for vector maximization with respect to cones. Journal of Mathematical Analysis and Applications [J],1979, 71(1):232-241.
[16]M.I. Henig. Proper efficiency with respect to cones. Journal of Optimization Theory and Applications [J],1982,36(3):387-407.
[17]J.M. Borwein and D.M. Zhuang. Super efficiency in convex vector optimization. Mathematical Methods of Operations Research[J],1991,35(3):175-184.
[18]J.M. Borwein and D.M. Zhuang. Super efficiency in vector optimization. Trans-actions of the American Mathematical Society[J],1993,338:105-122.
[19]A. Guerraggio, E. Molho and A. Zaffaroni. La nozione di punto propriamente pareto-efficiente, Technical Paper No.56, Istituto di Matematica Finanziaria, Uni-versit di Torino, Torino, Italy,1990.
[20]A. Guerraggio, E. Molho and A. Zaffaroni. On the notion of proper efficiency in vector optimization. Journal of Optimization Theory and Applications [J],1994, 82(1):1-21.
[21]杨新民Benson真有效解与Borwein真有效解的等价性.应用数学[J],1994,7(2)：246-247.
[22]X.Y. Zheng. Proper efficiency in locally convex topological vector spaces. Journal of Optimization Theory and Applications [J],1997,94(2):469-486.
[23]D.E. Ward and G.M. Lee. Generalized properly efficient solutions of vector opti-mization problems. Mathematical Methods of Operations Research[J],2001,53(2): 215-232.
[24]H.P. Benson. Existence of efficient solutions for vector maximization problems. Journal of Optimization Theory and Applications[J],1978,26(4):569-580.
[25]J.M. Borwein. On the existence of pareto efficient points. Mathematics of Opera-tions Research[J],1983,8(1):64-73.
[26]D.T. Luc. An existence theorem in vector optimization. Mathematics of Opera-tions Research[J],1989,14(4):693-699.
[27]T.X. Duc Ha. On the existence of efficient points in locally convex spaces. Journal of Global Optimization [J],1994,4(3):265-278.
[28]Y. Sonntag and C. Zalinescu. Comparison of existence results for efficient points. Journal of Optimization Theory and Applications [J],2000,105(1):161-188.
[29]X.X. Huang and X.M. Yang. Some existence results of efficiency in vector opti-mization. Mathematical Methods of Operations Research[J],2001,53(3):391-401.
[30]K.F. Ng and X.Y. Zheng. Existence of efficient points in vector optimization and generalized bishop-phelps theorem. Journal of Optimization Theory and Applica-tions[J],2002,115(1):29-47.
[31]X.Y. Zheng, X.M. Yang and K.L. Teo. Super efficiency of vector optimization in Banach spaces. Journal of Mathematical Analysis and Applications [J],2007,327(1): 453-460.
[32]M. Oveisiha and J. Zafarani. Super efficient solutions for set-valued maps. Opti-mization[J], DOI:10.1080/02331934.2012.712119,2012.
[33]E.U. Choo and D.R. Atkins. Proper efficiency in nonconvex multicriteria program-ming. Mathematics of Operations Research[J],1983,8(3):467-470.
[34]D.T. Luc. Scalarization of vector optimization problems. Journal of Optimization Theory and Applications [J],1987,55(1):85-102.
[35]Z.F. Li and S.Y. Wang. Lagrange multipliers and saddle points in multiobjective programming. Journal of Optimization Theory and Applications [J],1994,83(1): 63-81.
[36]X.M. Yang. Generalized subconvexlike functions and multiple objective optimiza-tion. Systems Science and Mathematical Sciences[J],1995,8(3):254-259.
[37]Z.F. Li and G.Y. Chen. Lagrangian multipliers, saddle points and duality in vector optimization of set-valued maps. Journal of Mathematical Analysis and Applications[J],1997,215(2):297-316.
[38]Z.F. Li. Benson proper efficiency in the vector optimization of set-valued maps. Journal of Optimization Theory and Applications[J],1998,98(3):623-649.
[39]G.Y. Chen and W.D. Rong. Characterizations of the Benson proper efficiency for nonconvex vector optimization. Journal of Optimization Theory and Applica-tions[J],1998,98(2):365-384.
[40]W.D. Rong and Y.N. Wu. Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps. Mathematical Methods of Operations Research[J],1998,48(2):247-258.
[41]X.M. Yang, X.Q. Yang and G.Y. Chen. Theorems of the alternative and optimiza-tion with set-valued maps. Journal of Optimization Theory and Applications [J], 2000,107(3):627-640.
[42]X.Y. Zheng. Scalarization of Henig proper efficient points in a normed space. Journal of Optimization Theory and Applications [J],2000,105(1):233-247.
[43]X.M. Yang, D. Li and S.Y. Wang. Near-subconvexlikeness in vector optimization with set-valued functions. Journal of Optimization Theory and Applications [J], 2001,110(2):413-427.
[44]A. Mehra. Super efficiency in vector optimization with nearly convexlike set-valued maps. Journal of Mathematical Analysis and Applications [J],2002,276(2): 815-832.
[45]P.H. Sach. Nearly subconvexlike set-valued maps and vector optimization prob-lems. Journal of Optimization Theory and Applications [J],2003,119(2):335-356.
[46]A. Zaffaroni. Degrees of efficiency and degrees of minimality. SIAM Journal on Control and Optimization[J],2003,42(3):1071-1086.
[47]M. Adan and V. Novo. Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness. European Journal of Operational Re-search[J],2003,149(3):641-653.
[48]M. Ehrgott and M.M. Wiecek. Saddle points and pareto points in multiple objec-tive programming. Journal of Global Optimization[J],2005,32(1):11-33.
[49]D.T. Luc, T.Q. Phong and M. Volle. Scalarizing functions for generating the weakly efficient solution set in convex multiobjective problems. SIAM Journal on Optimization[J],2005,15(4):987-1001.
[50]X.Y. Zheng and X.Q. Yang. Lagrange multipliers in nonsmooth semi-infinite opti-mization problems. Mathematics of Operations Research[J],2007,32(1):168-181.
[51]L.Y. Xia and J.H. Qiu. Super efficiency in vector optimization with nearly sub-convexlike set-valued maps. Journal of Optimization Theory and Applications[J], 2008,136(1):125-137.
[52]J.H. Qiu. Dual characterization and scalarization for Benson proper efficiency. SIAM Journal on Optimization [J],2008,19(1):144-162.
[53]R. Kasimbeyli. A nonlinear cone separation theorem and scalarization in noncon-vex vector optimization. SIAM Journal on Optimization[J],2010,20(3):1591-1619.
[54]D.I. Duca and L. Lupsa. Saddle points for vector valued functions:existence, necessary and sufficient theorems. Journal of Global Optimization [J],2012,53(3): 431-440.
[55]Z.A. Zhou and X.M. Yang. Optimality conditions of generalized subconvexlike set-valued optimization problems based on the quasi-relative interior. Journal of Optimization Theory and Applications [J],2011,150(2):327-340.
[56]余国林.生成锥内部凸-锥-类凸集值优化问题的标量化和鞍点.应用泛函分析学报[J],2012,14(2)：206-212.
[57]W.T. Fu. The connectedness of the solution set of the multiobjective optimization problem for a set valued mapping. Applied Mathematics A Journal of Chinese Universities[J],1994,9(3):321-328.
[58]Z.F. Li and S.Y. Wang. Connectedness of super efficient sets in vector optimization of set-valued maps. Mathematical Methods of Operations Research[J],1998,48(2): 207-217.
[59]X.Y. Zheng. Contractibility and connectedness of efficient point sets. Journal of Optimization Theory and Applications[J],2000,104(3):717-737.
[60]M. Yu, S.Y. Wang, W.T. Fu and W.S. Xiao. On the existence and connectedness of solution sets of vector variational inequalities. Mathematical Methods of Operations Research[J],2001,54(2):201-215.
[61]X.H. Gong. On the contractibility and connectedness of an efficient point set. Journal of Mathematical Analysis and Applications [J],2001,264(2):465-478.
[62]凌晨.锥上半连续锥拟凸集值映射多目标优化超有效解集的连通性.系统科学与数学[J],2003,23(2)：163-169.
[63]Q.S. Qiu and X.M. Yang. Connectedness of Henig weakly efficient solution set for set-valued optimization problems. Journal of Optimization Theory and Appli-cations[J],2012,152(2):439-449.
[64]P.H. Naccache. Stability in vector optimization. Journal of Mathematical Analysis and Applications[J],1979,68:441-453.
[65]T. Tanino and Y. Sawaragi. Stability of nondominated solutions in multicriteria decision-making. Joural of Optimization Theory and Applcaiotns[J],1980,30(2): 229-253.
[66]D.T. Luc, R. Lucchetti and C. Maliverti. Convergence of the efficient sets. Set-Valued Analysis[J],1994,2(1-2):207-218.
[67]X.X. Huang. Stability in vector-valued and set-valued optimization. Mathematical Methods of Operations Research[J],2000,52(2):185-193.
[68]E. Miglierina and E. Molho. Convergence of minimal sets in convex vector opti-mization. SIAM Journal on Optimization[J],2005,15(2):513-526.
[69]S.W. Xiang and W.S. Yin. Stability results for efficient solutions of vector op-timization problems. Journal of Optimization Theory and Applications [J],2007, 134(3):385-398.
[70]C.S. Lalitha and P. Chatterjee. Stability and scalarization of weak efficient, effi-cient and henig proper efficient sets using generalized quasiconvexities. Journal of Optimization Theory and Applications [J],2012,155(3):941-961.
[71]H. Yano and M. Sakawa. A unified approach for characterizing Pareto optimal solu-tions of multiobjective optimization problems:The hyperplane method. European Journal of Operational Research[J],1989,39(1):61-70.
[72]P. Ruiz-Canales and A. Rufian-Lizana. A characterization of weakly efficient points. Mathematical Programming[J],1995,68(1-3):205-212.
[73]D.T. Luc and S. Schaible. Efficiency and generalized concavity. Joural of Opti-mization Theory and Applcaiotns[J],1997,94(1):147-153.
[74]S. Deng. Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization. Journal of Optimization Theory and Applications [J], 1998,96(1):123-131.
[75]W. Liu and X.H. Gong. Proper efficiency for set-valued vector optimization prob-lems and vector variational inequalities. Mathematical Methods of Operations Re-search[J],2000,51(3):443-457.
[76]X.X. Huang and X.Q. Yang. On characterizations of proper efficiency for noncon-vex multiobjective optimization. Journal of Global Optimization[J],2002,23(3-4): 213-231.
[77]X.X. Huang, X.Q. Yang and K.L. Teo. Characterizing nonemptiness and com-pactness of the solution set of a convex vector optimization problem with cone constraints and applications. Journal of Optimization Theory and Applications [J], 2004,123(2):391-407.
[78]P.H. Sach. Hartley proper efficiency in multiobjective optimization problems with locally lipschitz set-valued objectives and constraints. Journal of Global Optimiza-tion[J],2006,35(1):1-25.
[79]X.M. Yang and X.Q. Yang. Vector variational-like inequality with pseudoinvexity. Optimization [J],2006,55(1-2):157-170.
[80]F. Flores-Bazan and C. Vera. Characterization of the nonemptiness and com-pactness of solution sets in convex and nonconvex vector optimization. Journal of Optimization Theory and Applications[J],2006,130(2):185-207.
[81]X.Y. Zheng, X.M. Yang and K.L. Teo. Some geometrical aspects of efficient points in vector optimization. Journal of Optimization Theory and Applications [J],2007, 133(2):275-288.
[82]M. Chinaie and J. Zafarani. Image space analysis and scalarization of multivalued optimization. Journal of Optimization Theory and Applications [J],2009,142(3): 451-467.
[83]S. Deng. Characterizations of the nonemptiness and boundedness of weakly effi-cient solution sets of convex vector optimization problems in real reflexive Banach spaces. Journal of Optimization Theory and Applications [J],2009,140(1):1-7.
[84]S.S. Kutateladze. Convex ε-programming. Soviet Math Dokl[J],1979,20(2):391-393.
[85]P. Loridan. ε-solutions in vector minimization problems. Journal of Optimization Theory and Applications [J],1984,43(2):265-276.
[86]D.J. White. Epsilon efficiency. Journal of Optimization Theory and Applica-tions[J],1986,49(2):319-337.
[87]I. Valyi. Approximate saddle-point theorems in vector optimization. Joural of Optimization Theory and Applcaiotns[J],1987,55(3):435-448.
[88]S. Deng. On approximate solutions in convex vector optimization. SI AM Journal on Control and Optimization [J],1997,35(6):2128-2136.
[89]W.D. Rong and Y.N. Wu. e-Weak minimal solutions of vector optimization prob-lems with set-valued maps. Journal of Optimization Theory and Applications[J], 2000,106(3):569-579.
[90]J. Dutta and V. Vetrivel. On approximate minima in vector optimization. Nu-merical Functional Analysis and Optimization[J],2001,22(7-8):845-859.
[91]F.Y. Ren and W.D. Rong. Topological properties of ε-weak efficient sets in vector optimization with set-value maps. OR Transactions [J],2002,6(2):36-42.
[92]X.Q. Yang and X.Y. Zheng. Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces. Journal of Global Optimization [J], 2008,40(1-3):455-462.
[93]M. Chinaie and J. Zafarani. Image space analysis and scalarization for e-optimization of multifunctions. Journal of Optimization Theory and Applications [J], DOI:10.1007/ s10957-010-9657-6,2010.
[94]Q.S. Qiu and X.M. Yang. Some properties of approximate solutions for vector optimization problem with set-valued functions. Journal of Global Optimization[J], 2010,47(1):1-12.
[95]高英.多目标优化ε—拟弱有效解的最优性条件.应用数学学报.2010,33(6)：1061-1071.
[96]C.P. Liu, H.W.J. Lee and X.M. Yang. Optimality conditions and duality on ap-proximate solutions in vector optimization with arcwise connectivity. Optimization Letters[J],2012,6(8):1613-1626.
[97]Y. Gao, S.H. Hou and X.M. Yang. Existence and optimality conditions for approx-imate solutions to vector optimization problems. Journal of Optimization Theory and Applications[J],2012,152(1):97-120.
[98]C.P. Liu and H.W. Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial and Management Optimization [J],2012, 8(3):749-764.
[99]C. Gutierrez, B. Jimenez and V. Novo. A unified approach and optimality condi-tions for approximate solutions of vector optimization problems. SIAM Journal on Optimization[J],2006,17(3):688-710.
[100]C. Gutierrez, B. Jimenez and V. Novo. On approximate efficiency in multiobjec-tive programming. Mathematical Methods of Operations Research[J],2006,64(1): 165-185.
[101]C. Gutierrez, B. Jimenez and V. Novo. Optimality conditions via scalarization for a new ε-efficiency concept in vector optimization problems. European Journal of Operational Research[J],2010,201(1):11-22.
[102]M. Chicco, F. Mignanego, L. Pusillo and S. Tijs. Vector optimization problems via improvement sets. Journal of Optimization Theory and Applications [J],2011, 150(3):516-529.
[103]C. Gutierrez, B. Jimenez and V. Novo. Improvement sets and vector optimization. European Journal of Operational Research[J],2012,223(2):304-311.
[104]J.C. Liu. e-Properly efficient solutions to nondifferentiable multiobjective pro-gramming problems. Applied Mathematics Letters [J],1999,12(6):109-113.
[105]W.D. Rong and Y. Ma. ε-Properly efficient solutions of vector optimization problems with set-valued maps. OR Transactions[J],2000,4:21-32.
[106]C. Ling, ε-supper efficient solutions of vector optimization problems with set-value maps. OR Transactions[J],2001,5(3):51-56.
[107]C. Ling, ε-supper saddle points and ε-duality theorems of vector optimization problems with set-value maps. OR Transactions [J],2002,6(1):53-60.
[108]B.A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-damaneh. Scalarization for characterization of approximate strong/weak/proper efficiency in multi-objective optimization. Optimization[J], DOI:10.1080/02331934.2012.668190,2012.
[109]Y. Gao, X.M. Yang and K.L. Teo. Optimality conditions for approximate so-lutions of vector optimization problems. Journal of Industrial and Management Optimization[J],2011,7(2):483-496.
[110]C. Gutierrez, L. Huerga and V. Novo. Scalarization and saddle points of ap-proximate proper solutions in nearly subconvexlike vector optimization problems. Journal of Mathematical Analysis and Applications[J],2012,389(2):1046-1058.
[111]F. Plastria and E. Carrizosa. Geometrical characterization of weakly efficient points. Joural of Optimization Theory and Applcaiotns[J],1996,90(1):217-223.
[112]K. Winkler. Characterizations of efficient points in convex vector optimization problems. Mathematical Methods of Operations Research[J],2001,53(2):205-214.
[113]K. Winkler. A characterization of efficient and weakly efficient points in convex vector optimzation. SIAM Journal on Optimization[J],2008,19(1):756-765.
[114]T. Maeda. Constraint qualifications in multiobjective optimization problems: differentiable case. Joural of Optimization Theory and Applcaiotns[J],1994,80(3): 483-500.
[115]V. Preda and I. Chitescu. On constraint qualification in multiobjective opti-mization problems:semidifferentiable case. Journal of Optimization Theory and Applications[J],1999,100(2):417-433.
[116]G. Bigi and M. Pappalardo. Regularity conditions in vector optimization. Joural of Optimization Theory and Applcaiotns[J],1999,102(1):83-96.
[117]X.F. Li. Constraint qualifications in nonsmooth multiobjective optimization. Joural of Optimization Theory and Applcaiotns[J],2000,106(2):373-398.
[118]B. Jimenez and V. Novo. Alternative theorems and necessary optimality con-ditions for directionally differentiable multiobjective programs. Journal of Convex Analysis [J],2002,9(1):97-116.
[119]B. Jimenez and V. Novo. Optimality conditions in directionally differentiable pareto problems with a set constraint via tangent cones. Numerical Functional Analysis and Optimization [J],2003,24(5-6):557-574.
[120]G. Giorgi, B. Jimenez and V. Novo. On constraint qualifications in direction-ally differentiable multiobjective optimization problems. RAIRO Operations Re-search[J],2004,38(3):255-274.
[121]X.F. Li and J.Z. Zhang. Strong Kuhn-Tucker type conditions in nonsmooth multiobjective optimization:locally lipschitz case. Joural of Optimization Theory and Applcaiotns[J],2005,127(2):367-388.
[122]G. Giorgi, B. Jimenez and V. Novo. Strong Kuhn-Tucker conditions and con-straint qualifications in locally Lipschitz multiobjective optimization problems. Top[J]. 2009,17(2):288-304.
[123]M.B. Asadi and M. Soleimani-damaneh. Infinite alternative theorems and nons-mooth constraint qualification conditions. Set-Valued and Variational Analysis[J], 2012,20(4),551-566.
[124]M. Golestani and S. Nobakhtian. Nonsmooth multiobjective programming and constraint qualifications. Optimization[J], DOI:10.1080/02331934.2012.679939, 2012.
[125]M. Golestani and S. Nobakhtian. Convexificators and strong Kuhn-Tucker con-ditions. Computers and Mathematics with Applications [J],2012,64(4):550-557.
[126]R.S. Burachik and M.M. Rizvi. On weak and strong Kuhn-Tucker conditions for smooth multiobjective optimization. Joural of Optimization Theory and Ap-plcaiotns[J],2012,155(2),477-491.
[127]O.L. Mangasarian. A simple characterization of solution sets of convex programs. Operations Research Letters[J],1988,7(1):21-26.
[128]J.V. Burke and M.C. Ferris. Characterization of solution sets of convex programs. Operations Research Letters[J],1991,10(1):57-60.
[129]V. Jeyakumar and X.Q. Yang. Convex composite multi-objective nonsmooth programming. Mathematical Programming[J],1993.59(1-3):325-343.
[130]Z.L. Wu and S.Y. Wu. Characterizations of the solution sets of convex pro-grams and variational inequality problems. Joural of Optimization Theory and Applcaiotns[J],2006,130(2):341-360.
[131]V. Jeyakumar and X.Q. Yang. On characterizing the solution sets of pseudolinear programs. Journal of Optimization Theory and Applications[J],1995,87(3):747-755.
[132]Q.H. Lu and D.L. Zhu. Some characterizations of locally lipschitz pseudolinear functions. Applied Mathematics [J],2005,18(2):272-278.
[133]X.M. Yang. On characterizing the solution sets of pseudoinvex extremum prob-lems. Journal of Optimization Theory and Applications [J],2009,140(3):537-542.
[134]J.P. Penot. Characterization of solution sets of quasiconvex programs. Journal of Optimization Theory and Applications [J],2003,117(3):627-636.
[135]I. Vsevold. Characterizations of the solution sets of generalized convex minimiza-tion problems. Serdica Mathematical Journal[J].2003,29(1):1-10.
[136]S.K. Mishra and K.K. Lai. On characterization of solution sets of nonsmooth pseudoinvex minimization problems. IEEE Computer Society [J],2009,2:739-741.
[137]M. Castellani and M. Giuli. A characterization of the solution set of pseudoconvex extremum problems. Journal of Convex Analysis [J],2012,19(1):113-123.
[138]V. Jeyakumar, G.M. Lee and N. Dinh. Lagrange multiplier conditions character-izing optimal solution sets of convex programs. Applied Mathematics Report AMR 02/6, available at http://old.maths.unsw.edu.au/applied/files/2002/amr02.6.pdf.
[139]V. Jeyakumar, G.M. Lee and N. Dinh. Lagrange multiplier conditions charac-terizing the optimal solution sets of cone-constrained convex programs. Joural of Optimization Theory and Applcaiotns[J],2004,123(1):83-103.
[140]N. Dinh, V. Jeyakumar and G.M. Lee. Lagrange multiplier characterizations of solution sets of constrained pseudolinear optimization problems. Optimization [J], 2006,55(3):241-250.
[141]C.S. Lalithaa and M. Mehta. Characterizations of solution sets of mathematical programs in terms of Lagrange multipliers. Optimization[J],2009,58(8):995-1007.
[142]V. Jeyakumar, G.M. Lee and N. Dinh. Characterizations of solution sets of convex vector minimization problems. European Journal of Operational Research[J],2006, 174(3):1380-1395.
[143]G. Debreu. Theory of value[M]. John Wiley and Sons, New York,1959.
[144]E. Miglierina and E. Molho. Sectionwise connected sets in vector optimization. Operations Research Letters[J],2009,37(4):295-298.
[145]F.H. Clarke. Optimization and nonsmooth analysis[M]. SIAM, New York,1983.
[146]M.S. Bazaraa, H.D. Sherali and C.M. Shetty. Nonlinear programming, theory and algorithms (Third edition) [M]. John Wiley and Sons, Canada,2006.
[147]J.P. Penot and P.H. Quang. Generalized convexity of functions and general-ized monotonicity of set-valued maps. Journal of Optimization Theory and Ap-plcaiotns[J],1997,92(2):343-356.
[148]L.Y. Fan, S.Y. Liu and S.P. Gao. Generalized monotonicity and convexity of non-differentiable functions. Journal of Mathematical Analysis and Applications[J], 2003,279(1):276-289.
[149]T. Weir and B. Mond. Preinvex functions in multiobjective optimization. Journal of Mathematical Analysis and Applications [J],1988,136(1):29-38.
[150]T. Weir and V.Jeyakumar. A class of nonconvex functions and mathematical programming. Bulletin of Australian Mathematical Society [J],1988,38:177-189.
[151]T. Jabarootian and J. Zafarani. Generalized invariant monotonicity and invexity of non-differentiable functions. Journal of Global Optimization [J],2006,36(4):537-564.
[152]L.Y. Fan. Weak quasi-invexity of nonsmooth functions. Applied Mathematics Letters[J],2008,21(11):1112-1117.
[153]R. Pini. Invexity and generalized convexity. Optimization[J],1991,22(4):513-525.
[154]X.M. Yang, X.Q. Yang and K.L. Teo. Generalized invexity and generalized invari-ant monotonicity. Joural of Optimization Theory and Applcaiotns[J],2003,117(3): 607-625.
[155]S.R. Mohan and S.K. Neogy. On invex sets and preinvex functions. Journal of Mathematical Analysis and Applications[J],1995,189(3):901-908.
[156]X.M. Yang, X.Q. Yang and K.L. Teo. Criteria for generalized invex monotonici-ties. European Journal of Operational Research[J],2005,164(1):115-119.
[157]S.K. Mishra and G. Giorgi. Invexity and Optimization, Nonconvex Optimization and Its Applications [M]. Springer-Verlag, Berlin,2008.
[158]T. Tanaka and D. Kuroiwa, The convexity of A and B assures intA+B = int(A+B). Applied Mathematics Letters[J],1993,6(1):83-86.
[159]T. Tanaka and D. Kuroiwa. Another observation on conditions assuring intA+ B=int(A+B). Applied Mathematics Letters[J],1994,7(1):19-22.
[160]R.T. Rockafellar and R.J-B. Wets. Variational analysis[M]. Springer-Verlag, Berlin,2004.
[161]R.T. Rockafellar. Convex Analysis[M]. New Jersey:Princeton University Press, 1970.
[162]K.L. Chew and E.U. Choo. Pseudolinearity and efficiency. Mathematical Pro-gramming[J],1984,28(2):226-239.
[163]Q.H. Ansari, S. Schaible and J.C. Yao. η-Pseudolinearity. Riviste di Matematice per le Scienze Economiche e Sociali[J],1999,22(1-2):31-39.
[164]M.M. Makelaa and P. Neittaanmaki. nonsmooth optimization analysis and algo-rithms with applications to optimal control [M]. Word Scientific Publishing, Singa-pore,1992.
[165]M. Rezaie and J. Zafarani. Vector optimization and variational-like inequalities. Journal of Global Optimization[J],2009,43(1):47-66.
[166]Q.H. Ansari and M. Rezaei. Generalized pseudolinearity. Optimization Letters [J], 2012,6(2):241-251.