SHENG Bao-huai, LIU San-yang. Kuhn-Tucker Condition and the Wolfe Duality of Preinvex Set-Valued Optimization[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1447-1456.
Citation: SHENG Bao-huai, LIU San-yang. Kuhn-Tucker Condition and the Wolfe Duality of Preinvex Set-Valued Optimization[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1447-1456.

Kuhn-Tucker Condition and the Wolfe Duality of Preinvex Set-Valued Optimization

  • Received Date: 2004-09-17
  • Rev Recd Date: 2006-08-19
  • Publish Date: 2006-12-15
  • The optimality Kuhn-Tucker condition and the Wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied; Then, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.
  • loading
  • [1]
    Jahn J,Rauh R.Contingent epiderivative and set-valued optimization[J].Math Methods Oper Res,1997,46(2):193—211. doi: 10.1007/BF01217690
    Chen G Y,Jahn J.Optimality conditions for set-valued optimization problems[J].Math Methods Oper Res,1998,48(2):187—200. doi: 10.1007/s001860050021
    Yang X Q.Directional derivatives for set-valued mappings and applications[J].Math Methods Oper Res,1998,48(2):274—285.
    Jahn J,Khan A A.Generalized contingent epiderivatives in set-valued optimization: optimality conditions[J].Numberical Functional Analysis and Optimization,2002,23(7/8):807—831. doi: 10.1081/NFA-120016271
    Gtz A,Jahn J.The Lagrange multiplier rule in set-valued optimization[J].SIAM J Optim,1999,10(2):331—344.
    Huang Y W. Generalized constraint qualifications and optimality conditions for set-valued optimization problems[J].J Math Anal Appl,2002,265(2):309—321. doi: 10.1006/jmaa.2001.7705
    盛宝怀,刘三阳.Benson真有效意义下向量集值优化的广义Fritz John条件[J].应用数学和力学,2002,23(12):1289—1295.
    SHENG Bao-huai,LIU San-yang.The optimality conditions of nonconvex set-valued vector optimization[J].Acta Mathematica Scientia B,2002,22(1):47—55.
    Weir T,Mond B.Preinvex functions in multiple-objective optimization[J].J Math Anal Appl,1988,136(1):29—38. doi: 10.1016/0022-247X(88)90113-8
    Weir T,Jeyakumar V.A class of nonconvex functions and mathematical programming[J].Bull Austral Math Soc,1988,38(1):177—189. doi: 10.1017/S0004972700027441
    Yang X M, Yang X Q,Teo K L.Characterizations and applications of prequasi-invex functions[J].J Optim Theory Appl,2001,110(3):645—668. doi: 10.1023/A:1017544513305
    Mohan S R, Neogy S K. On invex sets and preinvex functions[J].J Math Anal Appl,1995,189(4):901—908. doi: 10.1006/jmaa.1995.1057
    Yang X M,Li Duan.On properties of preinvex functions[J].J Math Anal Appl,2001,256(2):229—241. doi: 10.1006/jmaa.2000.7310
    Yang X M,Li Duan. Semistrictly preinvex functions[J].J Math Anal Appl,2001,258(2):287—308. doi: 10.1006/jmaa.2000.7382
    Luo H Z, Xu Z K. On characterizations of prequasi-invex functions[J].J Optim Theory Appl,2004,120(2):429—439. doi: 10.1023/B:JOTA.0000015930.47489.b7
    Pini R.Invexity and generalized convexity[J].Optimization,1991,22(4):513—525. doi: 10.1080/02331939108843693
    Craven B D. Invex functions and constrained local minima[J].Bull Austral Math Soc,1981,24(2):357—366. doi: 10.1017/S0004972700004895
    Hanson M A. On sufficiency of the Kuhn-Tucker conditions[J].J Math Anal Appl,1981,80(3):545—550. doi: 10.1016/0022-247X(81)90123-2
    Suneja S K, Singh C,Bector C R.Generalization of preinvex and B-vex functions[J].J Optim Theory Appl,1993,76(3):577—587. doi: 10.1007/BF00939384
    Kaul R N, Kaur S. Optimality criteria in nonlinear programming involving nonconvex functions[J].J Math Anal Appl,1985,105(1):104—112. doi: 10.1016/0022-247X(85)90099-X
    Qsuna-G[KG-*5]. [KG-*5]. mez R,Beato-Moreno A,Rufian-lizana A.Generalized convexity in multiobjective programming[J].J Math Anal Appl,1999,233(2):205—220. doi: 10.1006/jmaa.1999.6284
    Mukherjee R N. Generalized pseudoconvex functions and multiobjective programming[J].J Math Anal Appl,1997,208(1):49—57. doi: 10.1006/jmaa.1997.5281
    Bhatia D, Mehra A.Lagrangian duality for preinvex set-valued functions[J].J Math Anal Appl,1997,214(3):599—612. doi: 10.1006/jmaa.1997.5599
    Yang X M, Li D,Wang S Y.Near-subconvexlikeness in vector optimization with set-valued functions[J].J Optim Theory Appl,2001,110(2):413—427. doi: 10.1023/A:1017535631418
    盛宝怀,刘三阳. 关于向量集值优化的Benson真有效性[J].应用数学,2000,13(4):95—99.
  • 加载中


    通讯作者: 陈斌,
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (2442) PDF downloads(905) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint