Duality and Weak Efficiency in Vector Variational Problems

We establish weak, strong, and converse duality results for weakly efficient solutions in vector or multiobjective variational problems, which extend and improve recent papers. For this purpose, we consider Kuhn–Tucker optimality conditions, weighting variational problems, and some classes of generalized convex functions, recently introduced, which are extended in this work. Furthermore, a related open question is discussed.

[1]  Manuel Arana-Jiménez,et al.  On variational problems: Characterization of solutions and duality , 2005 .

[2]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[3]  B. Mond,et al.  Duality and sufficiency in control problems with invexity , 1988 .

[4]  Manuel Arana-Jiménez,et al.  Pseudoinvexity, optimality conditions and efficiency in multiobjective problems; duality , 2008 .

[5]  Panos M. Pardalos,et al.  A survey of recent developments in multiobjective optimization , 2007, Ann. Oper. Res..

[6]  Panos M. Pardalos,et al.  Efficiency Conditions and Duality for a Class of Multiobjective Fractional Programming Problems , 2003, J. Glob. Optim..

[7]  Izhar Ahmad,et al.  Mixed type duality for multiobjective variational problems with generalized (F,ρ)-convexity , 2005 .

[8]  Laura Carosi,et al.  Recent Developments on Mathematical Programming and Applications , 2009 .

[9]  B. Mond,et al.  Generalized concavity and duality in continuous programming , 1984 .

[10]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[11]  M. A. Hanson,et al.  Bounds for functionally convex optimal control problems , 1964 .

[12]  Fernando M. Lobo Pereira,et al.  Control Design for Autonomous Vehicles: A Dynamic Optimization Perspective , 2001, Eur. J. Control.

[13]  A. M. Geoffrion.  Proper efficiency and the theory of vector maximization , 1968 .

[14]  B. Mond,et al.  Sufficient optimality criteria and duality for variational problems with generalised invexity , 1989, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[15]  M. A. Hanson.  On sufficiency of the Kuhn-Tucker conditions , 1981 .

[16]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[17]  Antonio Rufián-Lizana,et al.  Weak efficiency in multiobjective variational problems under generalized convexity , 2012, J. Glob. Optim..