集值广义向量变分不等式问题的研究
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摘要
集值映射广义向量变分不等值(GVVTI)是变分不等式的重要推广形式,是研究多目标规划、均衡问题以及其他数学和工程领域中等问题的重要理论基础和工具,对这一问题的研究涉及到集值分析、凸分析、线性与非线性分析、非光滑分析、泛函分析等,有重要的学术价值。本文主要从理论上研究Hausdorff拓扑向量空间上一类集值映射广义向量变分不等式解的存在性问题,通过添加下半连续、C-单调等条件,运用经典的KKM-Fan定理对其等价问题的解的存在性进行证明,这一结论也统一和推广了许多已有的向量变分不等式解的存在性问题的研究。
在求解变分不等式时,我们常把一个变分不等式问题转化为一个最优化问题,通过解决最优化问题从而达到求解变分不等式的目的,而Gap函数在这种转化过程中起到了至关重要的作用。Gap函数最初应用在优化问题上并得到了一系列较好的应用,但是把Gap函数运用到变分不等式问题上还是近几年发展起来的一种趋势。所以本文就所提出的Hausdorff拓扑向量空间集值广义向量变分不等式模型定义了Gap函数,并给出了此类集值广义向量变分不等式解存在的充分必要条件。
Generalize vector variational inequalities with set-valued mapping are very important generation of variational inequalities. They become significant foundation and tool for studying multiobjective program,equilibrium, as well as many problems in the fields of mathematics and engineering. Because generalized vector variational inequlitise touch upon many mathematical branches, such as applications in set-valued analysis,convex analysis, linear and nonlinear analysis,nonsmooth analysis, function analysis,the research for them is of much academic value. This dissertation is devoted to study the existence of solution of generalized vector variational inequalities with set-valued in Haudorff topological vector spaces by KKM-Fan theorem,through giving conditions such as lower semicontinuity and C-monotonicty. This result is also the unity and extension of a number of well-known existence theorem of solution of vector variational inequalities.
When solving variational inequalities, we often transform a variational inequality problem into an optimization problem. Then we can solve the variational inequality by solving the optimization problem. Gap function plays a crucial role in this kind of transformation. Gap function is always used in the optimization problem on the initial application and get a series of good application, but the Gap function applied to the variational inequality problem is developed in recent years as a trend. So this paper will define a kind of Gap function for the set-valued generalized vector variational inequality in the Hausdorff topological vector space and will established necessary and sufficient conditions for the existence of a solution for the GVVI.
在求解变分不等式时,我们常把一个变分不等式问题转化为一个最优化问题,通过解决最优化问题从而达到求解变分不等式的目的,而Gap函数在这种转化过程中起到了至关重要的作用。Gap函数最初应用在优化问题上并得到了一系列较好的应用,但是把Gap函数运用到变分不等式问题上还是近几年发展起来的一种趋势。所以本文就所提出的Hausdorff拓扑向量空间集值广义向量变分不等式模型定义了Gap函数,并给出了此类集值广义向量变分不等式解存在的充分必要条件。
Generalize vector variational inequalities with set-valued mapping are very important generation of variational inequalities. They become significant foundation and tool for studying multiobjective program,equilibrium, as well as many problems in the fields of mathematics and engineering. Because generalized vector variational inequlitise touch upon many mathematical branches, such as applications in set-valued analysis,convex analysis, linear and nonlinear analysis,nonsmooth analysis, function analysis,the research for them is of much academic value. This dissertation is devoted to study the existence of solution of generalized vector variational inequalities with set-valued in Haudorff topological vector spaces by KKM-Fan theorem,through giving conditions such as lower semicontinuity and C-monotonicty. This result is also the unity and extension of a number of well-known existence theorem of solution of vector variational inequalities.
When solving variational inequalities, we often transform a variational inequality problem into an optimization problem. Then we can solve the variational inequality by solving the optimization problem. Gap function plays a crucial role in this kind of transformation. Gap function is always used in the optimization problem on the initial application and get a series of good application, but the Gap function applied to the variational inequality problem is developed in recent years as a trend. So this paper will define a kind of Gap function for the set-valued generalized vector variational inequality in the Hausdorff topological vector space and will established necessary and sufficient conditions for the existence of a solution for the GVVI.
引文
[1]Chen, G. Y, Existence of solutions for a vertor variational inequality:an extension of Hartman-Stampacchia theorem, J. Optim. Theory Appl.445-456(1992)
[2]L. C. Ceng, S. Huang, Existence theorems for generalized vector variational inequality es with a variable ordering relation,J.Glob Optim.46,521-535(2010)
[3]Giannessi. F, Theorems of the alternative, quadratic programs and complementarity pro blem. In:Cottle, R. W., Giannessi, F., Lions, J. L. (eds.)Variational inequalities and comp lementarity problems,Wiley and Sons, New York (1980)
[4]Huang, N. J., Fang, Y. P., On vector variational inequalities in reflexive Banach space s.J. Glob>Optim.32,495-505(2005)
[5]Zeng, L. C., Yao, J. C., Existence of solutions of generalized vector variational inequa lities in reflexive Banach spaces. J. Glob. Optim.36,483-497(2006)
[6]Konnov, I.V., Yao, J. C.,On the generalized vector variational inequality problem. J. Ma th. Anal. Appl.206,42-58 (1997)
[7]Lin, K. L., Yang, D. P., Yao, J. C., Generalized vector variational inequalities. J. Optim. Th eory>Appl.92,117-125(1997)
[8]Fan, K.:A generalization of Tychonoff s fixed-point theorem. Math. Ann.142,305-310(1 961)
[9]Si-jia Jiang, Li-Ping Pang, Jie Shen, Existence of solutions of generalized vector va riational-type inequalities with set-valued mappings. Computers&Mathematics with Ap plications.59,1453-1461(2010)
[10]Hartman G J, Stampacchia G. On some nonlinear elliptic differential functional equations [J].Acta Math.1996,115:271-310
[11]Lions J L, Stampacchia G. Vatiational inequalities. Communications in pure and applied mathematics [J],1967,20:493-512
[12]Cottle R W,Giannessi F, Lions J L. Variational Inequalities:Theory and Applications [C]. Amsterdam,Holland:North-Holland,1981
[13]Yu S J, Yao J C. On the vector variational inequalities [J]. Journal of Optimization Theory and Applications,1996,89:749-769.
[14]Lee B S, Lee G M, Lee S J. Variational-type inequalities for (η,θ,δ) pseudomonotone-type set-valued mappings in nonreflexive Banach space [J]. Applied Mathematics Letters,2002,15:109-114
[15]He Y R. Stable pseudomonotone variational inequality in reflexive Banach spaces [J].Joural of Mathematical Analysis and Applications,2007,300:352-363.
[16]Pang J S. Asymmetric variational inequality problems over product sets applications and iterative methods [J]. Mathematical Programming,1985,31:206-219.
[17]Cohen G, Chaplais F. Nested monotony for variational inequalities over a product of spaces and convergence if iterative algorithms [J]. Journal of Optimization Theory and Applications,1988,29:360-390.
[18]Kassay G, Kolumban J. System of multi-valued variational inequalities [J]. Publ. Math. Debrecen,2000,56,185-195.
[19]Ansari QH, Yao J C. A fixed point theorem and its applications to a system id variational inequalities [J]. Bull. Anstral.Math. Soc,1999,59:422-433.
[20]Fang Z. A generalized vector variational inequality problem with a set-valued semimonotone mapping [J]. Nonlinear Analysis,2008,69:1824-1829.
[21]Zhao Y L, Xia Z Q, Wang M Z. Gap functions and existence results for generalized vector variational inequalities [J]. Journal of Mathematical Inequalities,2007,1:137-148.
[22]Kum S, Lee G M. Remarks in implicit vector variational inequalities [J]. Taiwanese Journal of Mathematics,2002,6:369-382.
[23]Lee G M, Kum S. On implicit vector variational inequalities[J]. Journal of Optimization Theory and Applicationas,2000,104:409-425.
[24]Fang Y P, Huang N J. A new system of variational inclusions with (H,η)-monotone operators in Hilbert spaces[J]. Computers and Mathematics with Applications, 2005,49:365-374
[25]Hai N X, Khanhh P Q. The solution existence of general variational inclusion problems [J]. Journal of Optimization Theory and Applications,2007,328:1268-1277.
[26]Lin L J, Tu C. The studies of sustems of variational inclusions problems and variational disinclusions problems with applications [J]. Nonlinear Analtsis,2008,69:1981-1998.
[27]Peng J W, Zhu D L. A system of variational inclusions with P-η-acctetive operators [J]. Journal of Mathematical Analysis and Applications,2008,216:198-209.
[28]Verma R U. Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A,η)-resolvent operator technique [J]. Applied Mathematics Letters,2006,19:1409-1413.
[29]Kazm K R, Khan F A. Iterative approximation of a solution of multi-valued variational-like inclusion in Banach space:a P-η- proximal-point mapping approach [J].Journal of Mathematical Analysis and Applications,2007,325:665-674.
[30]Huang N J, Fang Y P. On vector variational inequalities in reflexive Banach spaces [J]. Journal of Global Optimiazation,2005,32:495-505.
[31]Fang Y P, Huang N J. Existence results for systems of strong implicit vector variational inequalities [J].Acta Mathmatica Hungarica,2004,103(4):267-277.
[32]Noor M A. Auxiliary principle for generalized mixed variational-like inequalities[J]. Journal of Mathmatic Analysis and Applications,1997,215:78-85.
[33]Huang N J, Deng C X, Auxiliary principle and iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities [J]. Journal of Mathmatical Analysis and Applications,2001,256:345-359.
[34]Zhang S S. Set-valued variational inclusions in Banach spaces [J]. Journal of Mathmatical Analysis and Applications,2000,248:438-454.
[35]Chen G Y, Cheng G M. Vector variational inequality and vector optimization [J]. Lecture Notes in Economics and Mathmatical Systems, Springer Verlag, Nem York,1987,258:408-416.
[36]Huang N J, Fang Y P. On vector variational inequalities in reflexive Banach spaces [J]. Journal of Global Optimiazation,2005,32:495-505.
[37]Yang X Q,Chen G Y. A class of nonconvex functions and variational inequalities [J]. Journal of Mathmatical Analysis and Applications,1992,169:359-373.
[38]Yang X Q, Goh C J. On vector variational inequality application to vector traffic equilibria [J]. Journal of Mathmatical Analysis and Applications,1997,95:431-443.
[39]Yang X Q. Vector variational inequality and its duality [J]. Nonlinear Analysis,1993,21:869-877.
[40]Chen G Y, B D Craven. Approximate dual and approximate vector variational inequality for multiobjective optimization [J].Aust.Math. Soc. Series A,1989,47:418-423.
[41]E. BEHERA, L. NAYAK. On nonlinear variational-type inequality problem [J].Pure Appl.Math,1999,30(9):911-923.
[42]F. Giannessi. Vector variational inequalities and vector equilibrium, Kluwer Academic Publishers, Dordrecht, Boston, London,2000.
[43]Li J,Huang N J, Kim J M. On implicit vector equilibrium problems [J]. Journal of Mathmatical Analysis and Applications,2003,283:437-452.
[44]Yang X Q, Yao J C. Gap functions and existence of solutions to set-valued vector variational inequalities [J]. Journal of Mathmatical Analysis and Applications.2002,115:407-417.
[45]Li J, He Z Q. Gap functions and existence of solutions to generalized vector variational inequalities [J]. Applied Mathematics Letters.2005,18:989-1000.
[46]R. Reemtsen, J J Ruckmann, Editors. Semi-infinite Programming, Kluwer Acadmic Publishers, Dordrecht, Holland,1998.
[47]Chen G Y. Existence of solutions for a vector vatiational inequality:an extension of Hartmann-Stampacchia theorem [J]. Journal of Optimization Theory and Applications.1992,74:445-456.
[48]Fan K. A generalization of Tychonoffs fixed point theorem [J]. Math. Ann.1961,142:305-310
[49]Lee GM, Kum S H. On implicit vector variational inequalities [J]. Journal of Optimization Theory and Applications.2000,104(2):409-425.
[50]Lin K L, Yang D P, Yao J C. Generalized vector variational inequalities [J]. Journal of Optimization Theory and Applications.1997,92:117-125.
[51]A H Siddiqi, Q H Ansari.A Khaliq. On vector variational inequalities [J]. Journal of Optimization Theory and Applications.1995,84:171-180.
[52]Yu S J, Yao J C. On vector variational inequalities [J]. Journal of Optimization Theory and Applications.1996,89:749-769.
[53]Goh C J, Yang X Q. Vector equilibrium problems and vector optimization [J]. European Journal of Operational Research.1999,116:615-628.
[54]Browder F E, Hess P. Nonlinear mapping of monotone type in Banach spaces [J]. Journal of Functional Analysis.1972,11:251-294.
[55]Kravvarittis, D. Nonlinear equations and inequalities in Banach spaces [J]. Journal of Mathmatical Analysis and Applications.1979,67:205-214.
[56]Ansari Q H, Yao J C. Nondiffrerntiable and nonconvex optimization problems [J]. Journal of Optimization Theory and Applications.2000,106:475-488.
[57]Chen G Y, Gou C J, Yang X Q. On a gap function for vector variational inequalities, vector variational inequalities and vector equilibria,edited by F Giannessi, Kluwer Academic Publishers, Dordrecht, Holland.2000:55-72.
[58]Yang X Q, Yao J C. Gap functions and existence of solutions to set-valued vector variational inequalities [J]. Journal of Optimization Theory and Applications.2002,115:407-417.
[59]赵亚莉.广义似变分不等式解的存在性和算法,大连：大连理工大学数学科学学院,2006.
[60]姜思佳.集值广义向量变分不等式解的存在性,大连：大连理工大学数学科学学院,2009.
[61]肖刚.几类向量变分不等式的研究,西安：西安电子科技大学应用数学,2008.
[2]L. C. Ceng, S. Huang, Existence theorems for generalized vector variational inequality es with a variable ordering relation,J.Glob Optim.46,521-535(2010)
[3]Giannessi. F, Theorems of the alternative, quadratic programs and complementarity pro blem. In:Cottle, R. W., Giannessi, F., Lions, J. L. (eds.)Variational inequalities and comp lementarity problems,Wiley and Sons, New York (1980)
[4]Huang, N. J., Fang, Y. P., On vector variational inequalities in reflexive Banach space s.J. Glob>Optim.32,495-505(2005)
[5]Zeng, L. C., Yao, J. C., Existence of solutions of generalized vector variational inequa lities in reflexive Banach spaces. J. Glob. Optim.36,483-497(2006)
[6]Konnov, I.V., Yao, J. C.,On the generalized vector variational inequality problem. J. Ma th. Anal. Appl.206,42-58 (1997)
[7]Lin, K. L., Yang, D. P., Yao, J. C., Generalized vector variational inequalities. J. Optim. Th eory>Appl.92,117-125(1997)
[8]Fan, K.:A generalization of Tychonoff s fixed-point theorem. Math. Ann.142,305-310(1 961)
[9]Si-jia Jiang, Li-Ping Pang, Jie Shen, Existence of solutions of generalized vector va riational-type inequalities with set-valued mappings. Computers&Mathematics with Ap plications.59,1453-1461(2010)
[10]Hartman G J, Stampacchia G. On some nonlinear elliptic differential functional equations [J].Acta Math.1996,115:271-310
[11]Lions J L, Stampacchia G. Vatiational inequalities. Communications in pure and applied mathematics [J],1967,20:493-512
[12]Cottle R W,Giannessi F, Lions J L. Variational Inequalities:Theory and Applications [C]. Amsterdam,Holland:North-Holland,1981
[13]Yu S J, Yao J C. On the vector variational inequalities [J]. Journal of Optimization Theory and Applications,1996,89:749-769.
[14]Lee B S, Lee G M, Lee S J. Variational-type inequalities for (η,θ,δ) pseudomonotone-type set-valued mappings in nonreflexive Banach space [J]. Applied Mathematics Letters,2002,15:109-114
[15]He Y R. Stable pseudomonotone variational inequality in reflexive Banach spaces [J].Joural of Mathematical Analysis and Applications,2007,300:352-363.
[16]Pang J S. Asymmetric variational inequality problems over product sets applications and iterative methods [J]. Mathematical Programming,1985,31:206-219.
[17]Cohen G, Chaplais F. Nested monotony for variational inequalities over a product of spaces and convergence if iterative algorithms [J]. Journal of Optimization Theory and Applications,1988,29:360-390.
[18]Kassay G, Kolumban J. System of multi-valued variational inequalities [J]. Publ. Math. Debrecen,2000,56,185-195.
[19]Ansari QH, Yao J C. A fixed point theorem and its applications to a system id variational inequalities [J]. Bull. Anstral.Math. Soc,1999,59:422-433.
[20]Fang Z. A generalized vector variational inequality problem with a set-valued semimonotone mapping [J]. Nonlinear Analysis,2008,69:1824-1829.
[21]Zhao Y L, Xia Z Q, Wang M Z. Gap functions and existence results for generalized vector variational inequalities [J]. Journal of Mathematical Inequalities,2007,1:137-148.
[22]Kum S, Lee G M. Remarks in implicit vector variational inequalities [J]. Taiwanese Journal of Mathematics,2002,6:369-382.
[23]Lee G M, Kum S. On implicit vector variational inequalities[J]. Journal of Optimization Theory and Applicationas,2000,104:409-425.
[24]Fang Y P, Huang N J. A new system of variational inclusions with (H,η)-monotone operators in Hilbert spaces[J]. Computers and Mathematics with Applications, 2005,49:365-374
[25]Hai N X, Khanhh P Q. The solution existence of general variational inclusion problems [J]. Journal of Optimization Theory and Applications,2007,328:1268-1277.
[26]Lin L J, Tu C. The studies of sustems of variational inclusions problems and variational disinclusions problems with applications [J]. Nonlinear Analtsis,2008,69:1981-1998.
[27]Peng J W, Zhu D L. A system of variational inclusions with P-η-acctetive operators [J]. Journal of Mathematical Analysis and Applications,2008,216:198-209.
[28]Verma R U. Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A,η)-resolvent operator technique [J]. Applied Mathematics Letters,2006,19:1409-1413.
[29]Kazm K R, Khan F A. Iterative approximation of a solution of multi-valued variational-like inclusion in Banach space:a P-η- proximal-point mapping approach [J].Journal of Mathematical Analysis and Applications,2007,325:665-674.
[30]Huang N J, Fang Y P. On vector variational inequalities in reflexive Banach spaces [J]. Journal of Global Optimiazation,2005,32:495-505.
[31]Fang Y P, Huang N J. Existence results for systems of strong implicit vector variational inequalities [J].Acta Mathmatica Hungarica,2004,103(4):267-277.
[32]Noor M A. Auxiliary principle for generalized mixed variational-like inequalities[J]. Journal of Mathmatic Analysis and Applications,1997,215:78-85.
[33]Huang N J, Deng C X, Auxiliary principle and iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities [J]. Journal of Mathmatical Analysis and Applications,2001,256:345-359.
[34]Zhang S S. Set-valued variational inclusions in Banach spaces [J]. Journal of Mathmatical Analysis and Applications,2000,248:438-454.
[35]Chen G Y, Cheng G M. Vector variational inequality and vector optimization [J]. Lecture Notes in Economics and Mathmatical Systems, Springer Verlag, Nem York,1987,258:408-416.
[36]Huang N J, Fang Y P. On vector variational inequalities in reflexive Banach spaces [J]. Journal of Global Optimiazation,2005,32:495-505.
[37]Yang X Q,Chen G Y. A class of nonconvex functions and variational inequalities [J]. Journal of Mathmatical Analysis and Applications,1992,169:359-373.
[38]Yang X Q, Goh C J. On vector variational inequality application to vector traffic equilibria [J]. Journal of Mathmatical Analysis and Applications,1997,95:431-443.
[39]Yang X Q. Vector variational inequality and its duality [J]. Nonlinear Analysis,1993,21:869-877.
[40]Chen G Y, B D Craven. Approximate dual and approximate vector variational inequality for multiobjective optimization [J].Aust.Math. Soc. Series A,1989,47:418-423.
[41]E. BEHERA, L. NAYAK. On nonlinear variational-type inequality problem [J].Pure Appl.Math,1999,30(9):911-923.
[42]F. Giannessi. Vector variational inequalities and vector equilibrium, Kluwer Academic Publishers, Dordrecht, Boston, London,2000.
[43]Li J,Huang N J, Kim J M. On implicit vector equilibrium problems [J]. Journal of Mathmatical Analysis and Applications,2003,283:437-452.
[44]Yang X Q, Yao J C. Gap functions and existence of solutions to set-valued vector variational inequalities [J]. Journal of Mathmatical Analysis and Applications.2002,115:407-417.
[45]Li J, He Z Q. Gap functions and existence of solutions to generalized vector variational inequalities [J]. Applied Mathematics Letters.2005,18:989-1000.
[46]R. Reemtsen, J J Ruckmann, Editors. Semi-infinite Programming, Kluwer Acadmic Publishers, Dordrecht, Holland,1998.
[47]Chen G Y. Existence of solutions for a vector vatiational inequality:an extension of Hartmann-Stampacchia theorem [J]. Journal of Optimization Theory and Applications.1992,74:445-456.
[48]Fan K. A generalization of Tychonoffs fixed point theorem [J]. Math. Ann.1961,142:305-310
[49]Lee GM, Kum S H. On implicit vector variational inequalities [J]. Journal of Optimization Theory and Applications.2000,104(2):409-425.
[50]Lin K L, Yang D P, Yao J C. Generalized vector variational inequalities [J]. Journal of Optimization Theory and Applications.1997,92:117-125.
[51]A H Siddiqi, Q H Ansari.A Khaliq. On vector variational inequalities [J]. Journal of Optimization Theory and Applications.1995,84:171-180.
[52]Yu S J, Yao J C. On vector variational inequalities [J]. Journal of Optimization Theory and Applications.1996,89:749-769.
[53]Goh C J, Yang X Q. Vector equilibrium problems and vector optimization [J]. European Journal of Operational Research.1999,116:615-628.
[54]Browder F E, Hess P. Nonlinear mapping of monotone type in Banach spaces [J]. Journal of Functional Analysis.1972,11:251-294.
[55]Kravvarittis, D. Nonlinear equations and inequalities in Banach spaces [J]. Journal of Mathmatical Analysis and Applications.1979,67:205-214.
[56]Ansari Q H, Yao J C. Nondiffrerntiable and nonconvex optimization problems [J]. Journal of Optimization Theory and Applications.2000,106:475-488.
[57]Chen G Y, Gou C J, Yang X Q. On a gap function for vector variational inequalities, vector variational inequalities and vector equilibria,edited by F Giannessi, Kluwer Academic Publishers, Dordrecht, Holland.2000:55-72.
[58]Yang X Q, Yao J C. Gap functions and existence of solutions to set-valued vector variational inequalities [J]. Journal of Optimization Theory and Applications.2002,115:407-417.
[59]赵亚莉.广义似变分不等式解的存在性和算法,大连：大连理工大学数学科学学院,2006.
[60]姜思佳.集值广义向量变分不等式解的存在性,大连：大连理工大学数学科学学院,2009.
[61]肖刚.几类向量变分不等式的研究,西安：西安电子科技大学应用数学,2008.