拟凸多目标优化问题近似解的最优性条件
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摘要
研究了拟凸多目标优化问题近似弱有效解、近似有效解的最优性条件.首先,在已有拟凸函数次微分的基础上引进4种近似次微分的概念,并给出它们之间的关系.然后,将4种近似次微分的概念应用到拟凸多目标优化问题中,给出了拟凸多目标优化问题近似弱有效解和近似有效解的充分条件和必要条件,并给出实例加以说明.
In this paper, we study the optimality conditions of approximate weak efficient solutions and approximate efficient solutions for quasiconvex multiobjective optimization problems. We introduce four concepts of approximate subdifferentials based on the existing subdifferentials of quasiconvex function, and give the relationship among them. And then, we apply these concepts to the quasiconvex multiobjective optimization problems, derive the sufficient conditions and necessary conditions for the approximate weak efficient solutions and the approximate efficient solution, and give some examples to illustrate the main results.
In this paper, we study the optimality conditions of approximate weak efficient solutions and approximate efficient solutions for quasiconvex multiobjective optimization problems. We introduce four concepts of approximate subdifferentials based on the existing subdifferentials of quasiconvex function, and give the relationship among them. And then, we apply these concepts to the quasiconvex multiobjective optimization problems, derive the sufficient conditions and necessary conditions for the approximate weak efficient solutions and the approximate efficient solution, and give some examples to illustrate the main results.
引文
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[6]李飞,唐莉萍,杨新民.集值映射的一种锥凸性及标量化[J].运筹学学报,2016, 20(4):21-29.
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[8] Penot J P, Zalinescu C. Elements of quasiconvex subdifferential calculus[J]. Journal of Convex Analysis,2000, 7(7):243-269.
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[10] Plastria F. Lower subdifferentiable functions and their minimization by cutting planes[J].Journal of Optimization Theory and Applications, 1985, 46(1):37-53.
[11] Penot J P. What is quasiconvex analysis?[J]. Optimization, 2000, 47(1-2):35-110.
[12] Penot J P. Characterization of solution sets of quasiconvex programs[J]. Journal of Optimization Theory and Applications, 2003, 117(3):627-636.
[13] Nguyen T H L, Penot J P. Optimality conditions for quasiconvex programs[J]. Siam Journal on Optimization,2006, 17(2):500-510.
[14] Gao Y, Yang X M, Lee H W J. Optimality conditions for approximate solutions in multiobjective optimization problems[J]. Journal of Inequalities and Applications. 2010, 2010(1):620928.
[15] Suzuki S, Kuroiwa D. Optimality conditions and the basic constraint qualification for quasiconvex programming[J]. Nonlinear Analysis Theory Methods and Applications, 2011, 74(4):1279-1285.
[16] Khanh P Q, Quyen H T, Yao J C. Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials[J]. European Journal of Operational Research,2011, 212(2):235-241.
[17] Sawaragi Y, Nakayama H, Tanino T. Theory of Multiobjective Optimization[M]. New York:Academic Press, 1985.
[2] Duca D I, Lupa L. On the e-epigraph of an e-convex function[J]. Journal of Optimization Theory and Applications, 2006, 129(2):341-348.
[3]杨新民.拟凸函数的某些性质[J].工程数学学报,1993, 10(1):51-56.
[4] Yang X M, Liu S Y. Technical note three kind of generalized convexity[J]. Journal of Optimization Theory and Applications, 1995, 86(2):501-513.
[5]杨新民,戎卫东.广义凸性及其应用[M].北京:科学出版社,2016.
[6]李飞,唐莉萍,杨新民.集值映射的一种锥凸性及标量化[J].运筹学学报,2016, 20(4):21-29.
[7] Greenberg H P, Pierskalla W P. Quasi-conjugate functions and surrogate duality[J]. Cahiers Centreetudes Recherche Oper, 1973, 15:437-448.
[8] Penot J P, Zalinescu C. Elements of quasiconvex subdifferential calculus[J]. Journal of Convex Analysis,2000, 7(7):243-269.
[9] Guti'errez Diez J M. Infragradients and directions of decrease[J]. Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid. Revista, 1984, 78:523-532.
[10] Plastria F. Lower subdifferentiable functions and their minimization by cutting planes[J].Journal of Optimization Theory and Applications, 1985, 46(1):37-53.
[11] Penot J P. What is quasiconvex analysis?[J]. Optimization, 2000, 47(1-2):35-110.
[12] Penot J P. Characterization of solution sets of quasiconvex programs[J]. Journal of Optimization Theory and Applications, 2003, 117(3):627-636.
[13] Nguyen T H L, Penot J P. Optimality conditions for quasiconvex programs[J]. Siam Journal on Optimization,2006, 17(2):500-510.
[14] Gao Y, Yang X M, Lee H W J. Optimality conditions for approximate solutions in multiobjective optimization problems[J]. Journal of Inequalities and Applications. 2010, 2010(1):620928.
[15] Suzuki S, Kuroiwa D. Optimality conditions and the basic constraint qualification for quasiconvex programming[J]. Nonlinear Analysis Theory Methods and Applications, 2011, 74(4):1279-1285.
[16] Khanh P Q, Quyen H T, Yao J C. Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials[J]. European Journal of Operational Research,2011, 212(2):235-241.
[17] Sawaragi Y, Nakayama H, Tanino T. Theory of Multiobjective Optimization[M]. New York:Academic Press, 1985.