优化及相关问题的研究
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摘要
优化问题、均衡问题与相补问题密切相关，本文对这三类问题进行研究。全文共五章，具体内容如下：
在第一章，我们考虑星形向量优化问题的稳定性与适定性。我们首先引入了具有沿射线增性质的三类映像，讨论了沿射线增性质与星形向量优化解的关系。当目标函数具有沿射线增性质时，我们研究了星形向量优化的稳定性与Dentcheva和Helbig意义下和Huang意义下的适定性问题。
在第二章，我们研究集值向量优化的适定性问题。对集值向量优化引入Bed-narczuk意义下的广义适定性概念，称之为广义B-适定性。广义B-适定性可以看作是借助于集合的Hausdorff收敛并考虑扰动的适定性。为了研究集值向量优化的广义B-适定性，我们对集值向量优化问题定义了几类(H)-性质，这些(H)-性质可以看作是Miglierina和Molho意义下的(H)-性质的集值与扰动推广。在凸性假设下，我们研究了集值向量优化的广义B-适定性与(H)-性质的紧密关系。
在第三章，我们研究均衡问题可行集的最小元问题。我们首次给出了均衡问题可行集的定义，并对二元函数给出Z-条件的定义。在Z-条件及严格单调性假设下，我们得到：均衡问题、均衡问题可行集的最小元问题以及一个相关的优化问题等价。进一步，在额外的增长性条件下，我们还得到均衡问题的可行集是一个子格。
在第四章，我们研究具(S)_+-条件的向量均衡系统问题的可解性。我们给出了映像族的(S)_+-条件的定义，这种(S)_+-条件覆盖了现有的各种(S)_+-条件。运用经典的Kakutani-Fan-Glicksberg不动点定理证明了一类具(S)_+-条件的向量均衡系统问题解的存在性。
在第五章，我们研究向量相补问题的可行性与可解性的关系。我们首先证明了：在伪单调性假设下，向量相补问题严格可行则一定可解。然后利用上述结果进一步证明：齐次向量相补问题可行则一定可解。最后，我们在乘积空间中研究了向量相补问题的可行性与可解性的关系。
In this thesis, we study three related problems: optimization problems, equilibrium problems and complementarity problems. This thesis is divided into five chapters. It is organized as follow:
In Chapter 1, we consider the stability and well-posedness of star-shaped vector optimization problems. We introduce three classes of increasing-alongrays maps and investigate the relations between increasing-along-rays property and star-shaped vector optimization problems. We also study the stability and well-posedness issues in the senses of Dentcheva and Helbig, and Huang in star-shaped vector optimization problems associated with increasing-along-rays maps.
In Chapter 2, we consider the well-posedness issue of set-valued vector optimization problems. We introduce the concept of extended well-posedness in the sense of Bednarczuk for set-valued vector optimization problems, which is named extended B-well-posedness. This notion of well-posedness can be interpreted as some sort of well-posedness under perturbation in terms of Hausdorff set-convergence. To investigate the extended B-well-posedness, we generalize property (H) due to Miglierina and Molho to set-valued and perturbed case. Under a convexity assumption, we show that the extended B-well-posedness is closely related to property (H).
In Chapter 3, we consider the least element problem of the feasible set for an equilibrium problem. We introduce the concepts of a feasible set for an equilibrium problem and of a Z-condition for a bifunction. Under Z-condition and strict pseudomonotonicity assumptions, we establish the equivalences among the equilibrium problem, the least element problem of the feasible set, and a related optimization problem. With an additional growth condition, we further prove that the feasible set of an equilibrium problem is a sublattice.
In Chapter 4, we consider the solvability of a system of vector equilibrium problems With (S)_+-conditions. We introduce the concept of (S)_+-conditions for a family of maps, which covers the concepts of existing (S)_+-conditions. We establish some existence theorems for systems of vector equilibrium problems with (S)_+-conditions by using the Kakutani-Fan-Glicksberg fixed point theorem.
In Chapter 5, we consider the relations between the feasibility and solvability of a vector complementarity problem. We prove that the vector complementarity problem with a pseudomonotonicity assumption is solvable whenever it is strictly feasible. By using the previous results, we further show that the homogeneous vector complementarity problem is solvable whenever it is feasible. At last, we study the solvability of the feasible vector complementarity problem on product spaces.
在第一章，我们考虑星形向量优化问题的稳定性与适定性。我们首先引入了具有沿射线增性质的三类映像，讨论了沿射线增性质与星形向量优化解的关系。当目标函数具有沿射线增性质时，我们研究了星形向量优化的稳定性与Dentcheva和Helbig意义下和Huang意义下的适定性问题。
在第二章，我们研究集值向量优化的适定性问题。对集值向量优化引入Bed-narczuk意义下的广义适定性概念，称之为广义B-适定性。广义B-适定性可以看作是借助于集合的Hausdorff收敛并考虑扰动的适定性。为了研究集值向量优化的广义B-适定性，我们对集值向量优化问题定义了几类(H)-性质，这些(H)-性质可以看作是Miglierina和Molho意义下的(H)-性质的集值与扰动推广。在凸性假设下，我们研究了集值向量优化的广义B-适定性与(H)-性质的紧密关系。
在第三章，我们研究均衡问题可行集的最小元问题。我们首次给出了均衡问题可行集的定义，并对二元函数给出Z-条件的定义。在Z-条件及严格单调性假设下，我们得到：均衡问题、均衡问题可行集的最小元问题以及一个相关的优化问题等价。进一步，在额外的增长性条件下，我们还得到均衡问题的可行集是一个子格。
在第四章，我们研究具(S)_+-条件的向量均衡系统问题的可解性。我们给出了映像族的(S)_+-条件的定义，这种(S)_+-条件覆盖了现有的各种(S)_+-条件。运用经典的Kakutani-Fan-Glicksberg不动点定理证明了一类具(S)_+-条件的向量均衡系统问题解的存在性。
在第五章，我们研究向量相补问题的可行性与可解性的关系。我们首先证明了：在伪单调性假设下，向量相补问题严格可行则一定可解。然后利用上述结果进一步证明：齐次向量相补问题可行则一定可解。最后，我们在乘积空间中研究了向量相补问题的可行性与可解性的关系。
In this thesis, we study three related problems: optimization problems, equilibrium problems and complementarity problems. This thesis is divided into five chapters. It is organized as follow:
In Chapter 1, we consider the stability and well-posedness of star-shaped vector optimization problems. We introduce three classes of increasing-alongrays maps and investigate the relations between increasing-along-rays property and star-shaped vector optimization problems. We also study the stability and well-posedness issues in the senses of Dentcheva and Helbig, and Huang in star-shaped vector optimization problems associated with increasing-along-rays maps.
In Chapter 2, we consider the well-posedness issue of set-valued vector optimization problems. We introduce the concept of extended well-posedness in the sense of Bednarczuk for set-valued vector optimization problems, which is named extended B-well-posedness. This notion of well-posedness can be interpreted as some sort of well-posedness under perturbation in terms of Hausdorff set-convergence. To investigate the extended B-well-posedness, we generalize property (H) due to Miglierina and Molho to set-valued and perturbed case. Under a convexity assumption, we show that the extended B-well-posedness is closely related to property (H).
In Chapter 3, we consider the least element problem of the feasible set for an equilibrium problem. We introduce the concepts of a feasible set for an equilibrium problem and of a Z-condition for a bifunction. Under Z-condition and strict pseudomonotonicity assumptions, we establish the equivalences among the equilibrium problem, the least element problem of the feasible set, and a related optimization problem. With an additional growth condition, we further prove that the feasible set of an equilibrium problem is a sublattice.
In Chapter 4, we consider the solvability of a system of vector equilibrium problems With (S)_+-conditions. We introduce the concept of (S)_+-conditions for a family of maps, which covers the concepts of existing (S)_+-conditions. We establish some existence theorems for systems of vector equilibrium problems with (S)_+-conditions by using the Kakutani-Fan-Glicksberg fixed point theorem.
In Chapter 5, we consider the relations between the feasibility and solvability of a vector complementarity problem. We prove that the vector complementarity problem with a pseudomonotonicity assumption is solvable whenever it is strictly feasible. By using the previous results, we further show that the homogeneous vector complementarity problem is solvable whenever it is feasible. At last, we study the solvability of the feasible vector complementarity problem on product spaces.
引文
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[2] Ansari, Q.H. and Yao, J.C.(2000) Systems of generalized variational inequalities and their applications, Appl. Anal. 76(3-4), 203-217.
[3] Ansari, Q.H., Schaible, S. and Yao, J.C.(2000) System of vector equilibrium problems and its applications, J. Optim. Theory Appl. 107(3), 547-557.
[4] Ansari, Q.H., Schaible, S. and Yao, J.C.(2002) The system of generalized vector equilibrium problems with applications, J. Global Optim. 22, 3-16.
[5] Ansari, Q.H., Chan, W.K. and Yang, X.Q.(2004) The system of vector quasi-equilibrium problems with applications, J. Global Optim. 29(1), 45-57.
[6] Ansari, Q.H., Lai, T.C and Yao, J.C.(1999) On the equivalence of extended generalized complementarity and generalized least element problems, J. Optim. Theory Appl. 102, 277-288.
[7] Ansari, Q.H., Yang, X.Q. and Yao, J.C.(2001) Existence and duality of implicit vector variational problems, Numer. Funct. Anal. Optim. 22(7-8), 815-829.
[8] Aubin, J.P. and Frankowska, H.(1990) Set-Valued Analysis, Systems & Control: Foundations & Applications, Vol. 2, Birkhaser, Basel.
[9] Browder, F.E.(1970) Pseudo-monotone operators and direct method of the calculus of variations, Arch. Rational Mech. Anal. 38, 268-277.
[10] Browder, F.E.(1983) Fixed-point theory and nonlinear problems, Bull. Amer. Math. Soc. 9, 1-39.
[11] Bednarczuk, E.(1987) Well posedness of vector optimization problems. In: Jahn J, Krabs W (eds) Recent advances and historical development of vector opti- mization problems, Lecture Notes in Economics and Mathematical systems, Vol 294, Springer-Verlag, Berlin, pp. 51-61.
[12] Bednarczuk, E. (1994) An approach to well-posedness in vector optimization: consequences to stability, Parametric optimization, Control Cybernet. 23(1-2), 107-122.
[13] Bednarczuk, E.M.(1996) Some stability results for vector optimization problems in partially ordered topological vector spaces, in: V. Lakshmikantham (eds.), Proceedings of the First World Congress of Nonlinear Analysis, Walter de Gruyter, Berlin, pp. 2371-2382.
[14] Bednarczuk, E.M.(2002) Upper Holder continuity of minimal points, J. Convex Anal. 9, 327-335.
[15] Bednarczuk, E.M.(2004) Continuity of minimal points with applications to parametric multiple objective optimization, European J. Oper. Res. 157, 59-67.
[16] Bianchi, M. and Schaible, S.(1996) Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl. 90, 31-43.
[17] Bianchi, M., Hadjisavvas, N. and Schaible, S.(1997) Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl. 92(3), 527-542.
[18] Blum, E. and Oettli, W. (1994) From optimization and variational inequalities to equilibrium problems, Math. Student 63(1-4), 123-145.
[19] Chadli, O., Wong, N.C. and Yao, J.C.(2003) Equilibrium problems with applications to eigenvalue problems, J. Optim. Theory Appl. 117(2), 245-266.
[20] Chadli, O., Chiang, Y. and Huang, S.(2002) Topological pseudomonotonicity and vector equilibrium problems, J. Math. Anal. Appl. 270, 435-450.
[21] Chen, G.Y.(1992) Existence of solutions for a vector variational inequality: an extension of the Hartman-Stampacchia theorem, J. Optim. Theory Appl. 74, 445-456,
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