AM  Vol.6 No.9 , August 2015
Mixed Saddle Point and Its Equivalence with an Efficient Solution under Generalized (V, p)-Invexity
ABSTRACT
The purpose of this paper is to define the concept of mixed saddle point for a vector-valued Lagrangian of the non-smooth multiobjective vector-valued constrained optimization problem and establish the equivalence of the mixed saddle point and an efficient solution under generalized (V, p)-invexity assumptions.

Cite this paper
Kumar, A. and Garg, P. (2015) Mixed Saddle Point and Its Equivalence with an Efficient Solution under Generalized (V, p)-Invexity. Applied Mathematics, 6, 1630-1637. doi: 10.4236/am.2015.69145.
References
[1]   Jeyakumar, V. and Mond, B. (1992) On Generalised Convex Mathematical Programming. Journal of the Australian Mathematical Society Series B, 34, 43-53.
http://dx.doi.org/10.1017/S0334270000007372

[2]   Mishra, S.K. and Mukherjee, R.N. (1996) On Generalised Convex Multi-Objective Nonsmooth Programming. Journal of the Australian Mathematical Society Series B, 38, 140-148.
http://dx.doi.org/10.1017/S0334270000000515

[3]   Liu, J.C. (1996) Optimality and Duality for Generalized Fractional Programming Involving Nonsmooth Pseudoinvex Functions. Journal of Mathematical Analysis and Applications, 202, 667-685.
http://dx.doi.org/10.1006/jmaa.1996.0341

[4]   Jeyakumar, V. (1985) Strong and Weak Invexity in Mathematical Programming. Mathematical Methods of Operations Research, 55, 109-125.

[5]   Jeyakumar, V. (1988) Equivalence of Saddle-Points and Optima, and Duality for a Class of Non-Smooth Non-Covex Problems. Journal of Mathematical Analysis and Applications, 130, 334-344.
http://dx.doi.org/10.1016/0022-247X(88)90309-5

[6]   Bector, C.R. (1996) Wolfe-Type Duality Involving (B, η)-Invex Functions for a Minmax Programming Problem. Journal of Mathematical Analysis and Applications, 201, 114-127.
http://dx.doi.org/10.1006/jmaa.1996.0245

[7]   Kuk, H., Lee, G.M. and Kim, D.S. (1998) Nonsmooth Multiobjective Programs with V-p-Invexity. Indian Journal of Pure and Applied Mathematics, 29, 405-412.

[8]   Bhatia, D. and Garg, P.K. (1998) (V, p)-Invexity and Non-Smooth Multiobjective Programming. RAIRO-Operations Research, 32, 399-414.

[9]   Bhatia, G. (2008) Optimality and Mixed Saddle Point Criteria Multiobjective Optimization. Journal of Mathematical Analysis and Applications, 342, 135-145.
http://dx.doi.org/10.1016/j.jmaa.2007.11.042

[10]   Antczak, T. (2015) Saddle Point Criteria and Wolfe Duality in Nonsmooth Φ, p-Invex Vector Optimization Problems with Inequality and Equality Constraints. International Journal of Computer Mathematics, 92, 882-907.
http://dx.doi.org/10.1080/00207160.2014.925191

[11]   Gutierrez, C., Hueraga, L. and Novo, V. (2012) Scalarization and Saddle Points of Approximate Proper Solutions in Nearly Subconvexlike Vector Optimization Problems. Journal of Mathematical Analysis and Applications, 389, 1046-1058.
http://dx.doi.org/10.1016/j.jmaa.2011.12.050

[12]   Chen, G.-Y. (1997) Lagrangian Multipliers, Saddle Points, and Duality in Vector Optimization of Set-Valued Maps. Journal of Mathematical Analysis and Applications, 215, 297-316.
http://dx.doi.org/10.1006/jmaa.1997.5568

[13]   Arrow, K.J., Gould, F.J. and Howe, S.M. (1973) A General Saddle Point Result for Constrained Optimization. Mathematical Programming, 5, 225-234.
http://dx.doi.org/10.1007/BF01580123

[14]   Reddy, L.V. and Mukherjee, R.N. (1999) Composite Nonsmooth Multiobjective Programs with V-p-Invexity. Journal of Mathematical Analysis and Applications, 235, 567-577.
http://dx.doi.org/10.1006/jmaa.1999.6409

[15]   Yuan, D., Liu, X. and Lai, G. (2012) Nondifferentiable Mathematical Programming Involving (G, β)-Invexity. Journal of Inequalities and Applications, 256, 01-17.
http://dx.doi.org/10.1186/1029-242x-2012-256

 
 
Top