多目标规划问题的对偶理论研究
摘要
多目标规划问题是一类非常重要的优化问题，这是因为现实世界中大部分最优化问题都涉及多个目标。而多目标规划的研究中，对偶理论研究是其中一个重要的研究方向，它对多目标最优化问题的求解以及最优性条件的揭示等都起着重要的作用。同时，凸优化问题是一类十分重要且基础的优化问题，但大多数的优化问题均是非凸的，为满足实际问题的需要，许多学者对凸性作了多种形式的推广，其中G不变凸性是一类重要的推广形式。因此，在G不变凸性下研究多目标规划问题的对偶理论具有十分重要的理论意义和较强的应用价值。本文即是在G不变凸性下研究两类多目标规划问题的对偶理论。一类是非可微多目标规划问题，另一类是可微多目标半无限规划问题。
第一章介绍多目标规划对偶理论的研究意义和相关问题的研究现状。
第二章介绍全文所需要的一些预备知识，给出相关概念和模型。
第三章主要研究非可微G不变凸多目标规划问题的对偶理论。首先给出此类问题的G Karush-Kuhn-Tucker必要条件。其次，提出此类问题的Mond-Weir对偶模型（NMWD1）和（NMWD2），分别研究两种对偶模型与原问题之间的对偶结果。其中，（NMWD2）是对（NMWD1）的改进，通过对（NMWD1）约束条件的改变得到更好的对偶结果。再次，定义G拉格朗日函数和Wolfe对偶模型（NWD），证明了（NWD）与原问题之间的弱对偶、强对偶和逆对偶定理。特别地，逆对偶定理和其中一个弱对偶定理的证明是在G拉格朗日函数不变凸的假设下给出的。最后，相对于Mond-Weir对偶模型（NMWD1）和（NMWD2），给出混合对偶模型（NMD1）和（NMD2），同样对两类模型下的各类对偶结果进行研究。
第四章在G不变凸性假设下得到可微的多目标半无限规划问题的对偶理论。研究了三类对偶模型，Mond-Weir、Wolfe和混合对偶模型，分别对三类模型下的对偶定理进行证明。
第五章对全文作了简单总结并提出了一些有待进一步研究的问题。
本文的创新之处主要体现在第三章、第四章。
Multiobjective programming problems is a great important class of optimizationproblem, because most of the optimization problems in our real life are multiobjective.And in multiobjective programming research, the dual theory research is one of theimportant research directions. It plays an important role in searching the solution ofmultiobjective optimization and optimality condition. Moreover, convexprogramming problem is an important and basic class of optimization problem,however, many programs are nonconvex. So as to meet the demand of solvingpractical problems, many generalizations to convexity have been made, among whichthe G invexity is an important form. Therefore, it is significant to study the dualitytheorem of multiobjective programming problem under G invexity. This paper isconcerned with the duality theorem of two classes of multiobjective programmingproblems. One of them is nondifferentiable multiobjective programming, the other ismultiobjective semi-infinite programming.
The outline of the thesis is as follows.
In chapter1, we introduce the research significance of multiobjective dualitytheorem and current situation of related problems.
In chapter2, we present some preliminaries and models for the full article.
In chapter3, we focus on the duality theory for nondifferentiable G invexmultiobjective programs. First of all, we introduce G Karush-Kuhn-Tuckernecessary optimality condition for such problem. Second, the Mond-Weir dual models(NMWD1) and (NMWD2) are proposed, we study its dual results.(NMWD2)improve (NMWD1) through changing the constrained conditions of (NMWD1), sothat better results can be obtained. Third, we define G lagrange function and Wolfedual model, weak duality、strong duality and converse duality are proved. Especially,we establish the converse duality and one of the weak duality under invex assumptionwhich imposed on G lagrange function. At last, related to (NMWD1) and(NMWD2), we present mixed dual models (NMD1) and (NMD2). Also we obtain thecorresponding dual results these two models and its primal problem.
In chapter4, we give the dual results for differentiable multiobjective semi-infinite programming under G invex assumption. Also we proposed three dualmodels, Mond-Weir、Wolfe and mixed duality, the corresponding dual theorems areproved.
In chapter5, we give a summary of this paper and put forward some problemsfor further study.
The innovation of this thesis lies in the3-th,4-th chapter.
第一章介绍多目标规划对偶理论的研究意义和相关问题的研究现状。
第二章介绍全文所需要的一些预备知识，给出相关概念和模型。
第三章主要研究非可微G不变凸多目标规划问题的对偶理论。首先给出此类问题的G Karush-Kuhn-Tucker必要条件。其次，提出此类问题的Mond-Weir对偶模型（NMWD1）和（NMWD2），分别研究两种对偶模型与原问题之间的对偶结果。其中，（NMWD2）是对（NMWD1）的改进，通过对（NMWD1）约束条件的改变得到更好的对偶结果。再次，定义G拉格朗日函数和Wolfe对偶模型（NWD），证明了（NWD）与原问题之间的弱对偶、强对偶和逆对偶定理。特别地，逆对偶定理和其中一个弱对偶定理的证明是在G拉格朗日函数不变凸的假设下给出的。最后，相对于Mond-Weir对偶模型（NMWD1）和（NMWD2），给出混合对偶模型（NMD1）和（NMD2），同样对两类模型下的各类对偶结果进行研究。
第四章在G不变凸性假设下得到可微的多目标半无限规划问题的对偶理论。研究了三类对偶模型，Mond-Weir、Wolfe和混合对偶模型，分别对三类模型下的对偶定理进行证明。
第五章对全文作了简单总结并提出了一些有待进一步研究的问题。
本文的创新之处主要体现在第三章、第四章。
Multiobjective programming problems is a great important class of optimizationproblem, because most of the optimization problems in our real life are multiobjective.And in multiobjective programming research, the dual theory research is one of theimportant research directions. It plays an important role in searching the solution ofmultiobjective optimization and optimality condition. Moreover, convexprogramming problem is an important and basic class of optimization problem,however, many programs are nonconvex. So as to meet the demand of solvingpractical problems, many generalizations to convexity have been made, among whichthe G invexity is an important form. Therefore, it is significant to study the dualitytheorem of multiobjective programming problem under G invexity. This paper isconcerned with the duality theorem of two classes of multiobjective programmingproblems. One of them is nondifferentiable multiobjective programming, the other ismultiobjective semi-infinite programming.
The outline of the thesis is as follows.
In chapter1, we introduce the research significance of multiobjective dualitytheorem and current situation of related problems.
In chapter2, we present some preliminaries and models for the full article.
In chapter3, we focus on the duality theory for nondifferentiable G invexmultiobjective programs. First of all, we introduce G Karush-Kuhn-Tuckernecessary optimality condition for such problem. Second, the Mond-Weir dual models(NMWD1) and (NMWD2) are proposed, we study its dual results.(NMWD2)improve (NMWD1) through changing the constrained conditions of (NMWD1), sothat better results can be obtained. Third, we define G lagrange function and Wolfedual model, weak duality、strong duality and converse duality are proved. Especially,we establish the converse duality and one of the weak duality under invex assumptionwhich imposed on G lagrange function. At last, related to (NMWD1) and(NMWD2), we present mixed dual models (NMD1) and (NMD2). Also we obtain thecorresponding dual results these two models and its primal problem.
In chapter4, we give the dual results for differentiable multiobjective semi-infinite programming under G invex assumption. Also we proposed three dualmodels, Mond-Weir、Wolfe and mixed duality, the corresponding dual theorems areproved.
In chapter5, we give a summary of this paper and put forward some problemsfor further study.
The innovation of this thesis lies in the3-th,4-th chapter.
引文
[1] R. T. Rockafellar. Convex Analysis[M]. New Jersey: Princeton University Press，1969.
[2] M. A. Hanson. On sufficiency of the Kuhn-Tucker conditions[J]. Journal of MathematicalAnalysis and Applications，1981，80(2):545-550.
[3] B. D. Craven. Invex functions and constrained local minima[J]. Bulletin of the AustralianMathematical Society，1981，24:357-366.
[4] T. Antczak.(p，r)-invex sets and functions[J]. Journal of Mathematical Analysis and Applications,2001,263:355-379.
[5] M. A. Hanson, B. Mond. Necessary and sufficient conditions in constrained optimization[J].Mathematical Programming,1987,37:51-58.
[6] A. Ben-Israel, B. Mond. What is invexity?[J]. Journal of the Australian Mathematical Society,1986, Ser. B28:1-9.
[7] B. D. Craven, B. M. Glover. Invex functions and duality[J]. Journal of the AustralianMathematical Society,1985, Ser.A39:1-20.
[8] D. H. Martin. The essence of invexity[J]. Journal of Optimization Theory and Applications,1985,47:65-76.
[9] T. Antczak. New optimality conditions and duality results of G-type in differentiablemathematical programming[J]. Nonlinear Analysis,2007,66:1617-1632.
[10] T. Antczak.(p, r)-invexity in multiobjective programming[J]. European Journal of OperationalResearch,2004,152:72-87.
[11] L. Batista dos Santos, R. Osuna-Gomez, M. A. Rojas-Medar, A. Rufian-Lizana. Invexitygeneralized and weakly efficient solutions for some vectorial optimization problem in Banachspaces[J]. Numerical Functional Analysis and Optimization,2004.
[12] G. Giorgi, A.Guerraggio. The notion of invexity in vector optimization: smooth and nonsmoothcase.In: Crouzeix, J.P. et al.(eds.), Generalized Convexity, Generalized Monotonicity. KluwerAcademic Publishers,1998.
[13] V. Jeyakumar, B. Mond. On generalized convex mathematical programming[J]. Journal of theAustralian Mathematical Society,1992, Ser.B34:43-53.
[14] R. Osuna-Gómez, A. Rufián-Lizana, P. Ruiz-Canales. Invex functions and generalizedconvexity in multiobjective programming[J]. Journal of Optimization Theory and Applications,1998,98:651-661.
[15] C. Singh. Optimality conditions in multiobjective differentiable programming[J]. Journal ofOptimization Theory and Applications,1987,53:115-123.
[16] T. Antczak. On G-invex multiobjective programming. Part I. Optimality[J]. Journal of GlobalOptimization,2009,43(1):97-109.
[17] H. Nakayama. Duality theory in vector optimization: an overview, decision making withmultiple objectives[J]. Economics and Mathematical Systems,1989,337:86-93.
[18] T. Taninio, Y. Sawaragi. Duality theory inmultiobjective programming[J]. Journal ofOptimization Theory and Applications,1987,53:115-123.
[19] D. T. Luc. On duality theory in multiobjective programming[J]. Journal of Optimization Theoryand Applications,1984,43(4):557-582.
[20] T. Weir, B. Mond, B. D. Craven. On duality for weakly minimized vector valued optimizationproblems[J]. Optimization,1986,17:711–721.
[21] R. R. Egudo. Efficiency and generalized convex duality formultiobjective programs[J]. Journalof Mathematical Analysis and Applications,1989,138:84–94.
[22] R. N. Kaul, S. K. Suneja, M.K. Srivastava. Optimality criteria and duality in multiple objectiveoptimization involving generalized invexity[J]. Journal of Optimization Theory andApplications,1994,80:465-482.
[23] T. Weir, V. Jeyakumar. A class of nonconvex functions and mathematical programming[J].Bulletin of the Australian Mathematical Society,1981,38:177-189.
[24] C. R. Bector, M. K. Bector, A. Gill, C. Singh. Duality for vector valued B-invex programming,1994, In:Proceedings Fourth International Workshop,Pecz,Hungary, pp:358-373.
[25] T. Antczak. The notion of V–r-invexity in differentiable multiobjective programming[J].Applicable Analysis,2005,11:63-79.
[26] T. Antczak. On G-invex multiobjective programming. Part II. Duality[J]. Journal of GlobalOptimization,2009,43(1):111-140.
[27] B. Mond, A class of nondifferentiable mathematical programming problems[J]. Journal ofMathematical Analysis and Applications,1974,46:169-174.
[28] J. Zhang, B. Mond. Duality for a nondifferentiable programming problem[J]. Bulletin of theAustralian Mathematical Society,1997,55:29-44.
[29] X. M. Yang, K. L. Teo, X. Q. Yang. Duality for a Class of Nondifferentiable MultiobjectiveProgramming Problems[J].Journal of Mathematical Analysis and Applications,2000,252:999-1005.
[30] D. S. Kim, S. J. Kim, and M. H. Kim. Optimality and duality for a class of nondifferentiablemultiobjective fractional programming problems[J]. Journal of Optimization Theory andApplications,2006,129:131-146.
[31] D. S. Kim, K. D. Bae. Optimality conditions and duality for a class of nondifferentiablemultiobjective programming problems[J]. Taiwanese Journal of Mathematics,2009,13:789-804.
[32] H. J. Kim, Y. Y. Seo, D. S. Kim. Optimality conditions in nondifferentiable G-invexmultiobjective programming[J]. Journal of Inequalities and Applications,2010, DOI:10.1155/2010/172059.
[33] A. Haar. Uber lineare Ungleichunge[J]. Acta Mathematica,1924,2:1-14.
[34] E. Remes. Surlecalcul effective des polinomes d’approximation de Tchebycheff[J]. ComptesRendus Mathematique. Academie des Sciences. Pairs,1934,199:337-340.
[35] K. Glashoff and S. A. Gustafson. Einfuhrung in die linear optimization[M]. WissenschaftlicheBuchgesell-sehaft, Darmstadt,1978.
[36] K. Glashoff and S. A. Gustafson. Linear optimization and approximation[M]. Springer, Newyork-Heidelberg-Berlin,1983.
[37] E. J. Aderson and A. B. Philpott, eds. Infinite programming[C]. Economy and Mathematicssystems259,1985, Springer-Berlin-Heidelberg-NewYork-Tokyo.
[38] A. V. Fiacco and K. Q. Kortanek, eds. Semi-infinite programming and applications[C]. LectureNotes in Eeomony and Mathematies Systems259,1983， Springer-Berlin-Heidelberg-Newyork-Tokyo.
[39] R. Hettich，editor. Semi-infinite programming[C].Lecture Notes in Control and InformationSeience(15),1979, Springer, Berlin-Heidelberg-NewYork.
[40] R. Hettich, P. Zencke. Numerische Mthoden der Approximation and semi-infiniteoptimieruun[M]. Teubner Stuttgart,1982.
[41] M. A. Goberna, M. A. Lopez. Linear semi-infinite programming theory: an updated survey[J].European Journal of Operational Research,2002,143:390-405.
[42] H. Gunzel, H. Th. Jogen, O. Stein. Generalized semi-infinite programming: the symmetricreduction ansatz[J]. Optimization Letters,2008,2(3):415-424.
[43] M. Lopez, G. Stil. Semi-infinite programming[J]. European Journal of Operational Research,2007,180:491-518.
[44] A. Shapiro. Semi-infinite programming, duality, discretization and optimality condition[J].Optimization,2009,58(2):133-161.
[45] N. Dinh, B. S. Morukhovich, T. A. Nghia. Subdifferentials of value functions and optimalityconditions for DC and bilevel infinite and semi-infinite programs[J]. MathematicalProgramming,2010,123(1):101-138.
[46] N. Kanzi, S. Nobakhtian. Nonsmooth semi-infinite programming problems with mixedconstraints[J]. Journal of Mathematical Analysis and Applications,2009,351:170-181.
[47] N. Kanzi, S. Nobakhtian. Optimality conditions for non-smooth semi-infinite programming[J].Optimization,2010,59(5):717-727.
[48] S.K.Mishra, M.Jaiswal, Le Thi Hoai An. Duality for nonsmooth semi-infinite programmingproblems[J]. Optimization Letters,2012,6(2):261-271.
[49] N. Kanzi. Necessary optimality conditions for nonsmooth semi-infinite programmingproblems[J]. Journal of Global Optimization,2011,49(4):713-725.
[50] S. K. Mishra, M. Jaiswal, H. A. Le Thi. Nonsmooth semi-infinite programming problem usingLimiting subdifferentials[J]. Journal of Global Optimization,2011, DOI10.1007/s10898-011-9690-5.
[51]史树中.凸分析[M].上海：上海科学技术出版社,1990.
[52]林锉云,董加礼.多目标优化的方法与理论[M].吉林：吉林教育出版社,1992.
[53] F. H. Clarke. Optimization and Nonsmooth Analysis[M]. New York: John Wiley,1983.
[2] M. A. Hanson. On sufficiency of the Kuhn-Tucker conditions[J]. Journal of MathematicalAnalysis and Applications，1981，80(2):545-550.
[3] B. D. Craven. Invex functions and constrained local minima[J]. Bulletin of the AustralianMathematical Society，1981，24:357-366.
[4] T. Antczak.(p，r)-invex sets and functions[J]. Journal of Mathematical Analysis and Applications,2001,263:355-379.
[5] M. A. Hanson, B. Mond. Necessary and sufficient conditions in constrained optimization[J].Mathematical Programming,1987,37:51-58.
[6] A. Ben-Israel, B. Mond. What is invexity?[J]. Journal of the Australian Mathematical Society,1986, Ser. B28:1-9.
[7] B. D. Craven, B. M. Glover. Invex functions and duality[J]. Journal of the AustralianMathematical Society,1985, Ser.A39:1-20.
[8] D. H. Martin. The essence of invexity[J]. Journal of Optimization Theory and Applications,1985,47:65-76.
[9] T. Antczak. New optimality conditions and duality results of G-type in differentiablemathematical programming[J]. Nonlinear Analysis,2007,66:1617-1632.
[10] T. Antczak.(p, r)-invexity in multiobjective programming[J]. European Journal of OperationalResearch,2004,152:72-87.
[11] L. Batista dos Santos, R. Osuna-Gomez, M. A. Rojas-Medar, A. Rufian-Lizana. Invexitygeneralized and weakly efficient solutions for some vectorial optimization problem in Banachspaces[J]. Numerical Functional Analysis and Optimization,2004.
[12] G. Giorgi, A.Guerraggio. The notion of invexity in vector optimization: smooth and nonsmoothcase.In: Crouzeix, J.P. et al.(eds.), Generalized Convexity, Generalized Monotonicity. KluwerAcademic Publishers,1998.
[13] V. Jeyakumar, B. Mond. On generalized convex mathematical programming[J]. Journal of theAustralian Mathematical Society,1992, Ser.B34:43-53.
[14] R. Osuna-Gómez, A. Rufián-Lizana, P. Ruiz-Canales. Invex functions and generalizedconvexity in multiobjective programming[J]. Journal of Optimization Theory and Applications,1998,98:651-661.
[15] C. Singh. Optimality conditions in multiobjective differentiable programming[J]. Journal ofOptimization Theory and Applications,1987,53:115-123.
[16] T. Antczak. On G-invex multiobjective programming. Part I. Optimality[J]. Journal of GlobalOptimization,2009,43(1):97-109.
[17] H. Nakayama. Duality theory in vector optimization: an overview, decision making withmultiple objectives[J]. Economics and Mathematical Systems,1989,337:86-93.
[18] T. Taninio, Y. Sawaragi. Duality theory inmultiobjective programming[J]. Journal ofOptimization Theory and Applications,1987,53:115-123.
[19] D. T. Luc. On duality theory in multiobjective programming[J]. Journal of Optimization Theoryand Applications,1984,43(4):557-582.
[20] T. Weir, B. Mond, B. D. Craven. On duality for weakly minimized vector valued optimizationproblems[J]. Optimization,1986,17:711–721.
[21] R. R. Egudo. Efficiency and generalized convex duality formultiobjective programs[J]. Journalof Mathematical Analysis and Applications,1989,138:84–94.
[22] R. N. Kaul, S. K. Suneja, M.K. Srivastava. Optimality criteria and duality in multiple objectiveoptimization involving generalized invexity[J]. Journal of Optimization Theory andApplications,1994,80:465-482.
[23] T. Weir, V. Jeyakumar. A class of nonconvex functions and mathematical programming[J].Bulletin of the Australian Mathematical Society,1981,38:177-189.
[24] C. R. Bector, M. K. Bector, A. Gill, C. Singh. Duality for vector valued B-invex programming,1994, In:Proceedings Fourth International Workshop,Pecz,Hungary, pp:358-373.
[25] T. Antczak. The notion of V–r-invexity in differentiable multiobjective programming[J].Applicable Analysis,2005,11:63-79.
[26] T. Antczak. On G-invex multiobjective programming. Part II. Duality[J]. Journal of GlobalOptimization,2009,43(1):111-140.
[27] B. Mond, A class of nondifferentiable mathematical programming problems[J]. Journal ofMathematical Analysis and Applications,1974,46:169-174.
[28] J. Zhang, B. Mond. Duality for a nondifferentiable programming problem[J]. Bulletin of theAustralian Mathematical Society,1997,55:29-44.
[29] X. M. Yang, K. L. Teo, X. Q. Yang. Duality for a Class of Nondifferentiable MultiobjectiveProgramming Problems[J].Journal of Mathematical Analysis and Applications,2000,252:999-1005.
[30] D. S. Kim, S. J. Kim, and M. H. Kim. Optimality and duality for a class of nondifferentiablemultiobjective fractional programming problems[J]. Journal of Optimization Theory andApplications,2006,129:131-146.
[31] D. S. Kim, K. D. Bae. Optimality conditions and duality for a class of nondifferentiablemultiobjective programming problems[J]. Taiwanese Journal of Mathematics,2009,13:789-804.
[32] H. J. Kim, Y. Y. Seo, D. S. Kim. Optimality conditions in nondifferentiable G-invexmultiobjective programming[J]. Journal of Inequalities and Applications,2010, DOI:10.1155/2010/172059.
[33] A. Haar. Uber lineare Ungleichunge[J]. Acta Mathematica,1924,2:1-14.
[34] E. Remes. Surlecalcul effective des polinomes d’approximation de Tchebycheff[J]. ComptesRendus Mathematique. Academie des Sciences. Pairs,1934,199:337-340.
[35] K. Glashoff and S. A. Gustafson. Einfuhrung in die linear optimization[M]. WissenschaftlicheBuchgesell-sehaft, Darmstadt,1978.
[36] K. Glashoff and S. A. Gustafson. Linear optimization and approximation[M]. Springer, Newyork-Heidelberg-Berlin,1983.
[37] E. J. Aderson and A. B. Philpott, eds. Infinite programming[C]. Economy and Mathematicssystems259,1985, Springer-Berlin-Heidelberg-NewYork-Tokyo.
[38] A. V. Fiacco and K. Q. Kortanek, eds. Semi-infinite programming and applications[C]. LectureNotes in Eeomony and Mathematies Systems259,1983， Springer-Berlin-Heidelberg-Newyork-Tokyo.
[39] R. Hettich，editor. Semi-infinite programming[C].Lecture Notes in Control and InformationSeience(15),1979, Springer, Berlin-Heidelberg-NewYork.
[40] R. Hettich, P. Zencke. Numerische Mthoden der Approximation and semi-infiniteoptimieruun[M]. Teubner Stuttgart,1982.
[41] M. A. Goberna, M. A. Lopez. Linear semi-infinite programming theory: an updated survey[J].European Journal of Operational Research,2002,143:390-405.
[42] H. Gunzel, H. Th. Jogen, O. Stein. Generalized semi-infinite programming: the symmetricreduction ansatz[J]. Optimization Letters,2008,2(3):415-424.
[43] M. Lopez, G. Stil. Semi-infinite programming[J]. European Journal of Operational Research,2007,180:491-518.
[44] A. Shapiro. Semi-infinite programming, duality, discretization and optimality condition[J].Optimization,2009,58(2):133-161.
[45] N. Dinh, B. S. Morukhovich, T. A. Nghia. Subdifferentials of value functions and optimalityconditions for DC and bilevel infinite and semi-infinite programs[J]. MathematicalProgramming,2010,123(1):101-138.
[46] N. Kanzi, S. Nobakhtian. Nonsmooth semi-infinite programming problems with mixedconstraints[J]. Journal of Mathematical Analysis and Applications,2009,351:170-181.
[47] N. Kanzi, S. Nobakhtian. Optimality conditions for non-smooth semi-infinite programming[J].Optimization,2010,59(5):717-727.
[48] S.K.Mishra, M.Jaiswal, Le Thi Hoai An. Duality for nonsmooth semi-infinite programmingproblems[J]. Optimization Letters,2012,6(2):261-271.
[49] N. Kanzi. Necessary optimality conditions for nonsmooth semi-infinite programmingproblems[J]. Journal of Global Optimization,2011,49(4):713-725.
[50] S. K. Mishra, M. Jaiswal, H. A. Le Thi. Nonsmooth semi-infinite programming problem usingLimiting subdifferentials[J]. Journal of Global Optimization,2011, DOI10.1007/s10898-011-9690-5.
[51]史树中.凸分析[M].上海：上海科学技术出版社,1990.
[52]林锉云,董加礼.多目标优化的方法与理论[M].吉林：吉林教育出版社,1992.
[53] F. H. Clarke. Optimization and Nonsmooth Analysis[M]. New York: John Wiley,1983.