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Optimality and duality for multiobjective fractional problems with \(r\)-invexity. (English) Zbl 1190.90191

Summary: A vector minimization problem involving nonlinear fractional functions is a natural extension of multiobjective linear fractional programming. Linear fractional criteria are frequently encountered in financial problem, game theory, decision theory, and all optimal decision problems with noncomparable criteria, e.g., in corporate planning and bank balance sheet management. Multiobjective (fractional) programming is indeed an interesting topic. Recently, there are many articles that have been of much interest, e.g., see [T. Antczak, Eur. J. Oper. Res. 137, No. 1, 28–36 (2002; Zbl 1027.90076); D. Bhatia and S. Pandey, Cah. Cent. Étud. Rech. Opér. 33, No. 1–2, 3–11 (1991; Zbl 0746.90073), D. Bhatia and P. K. Garg, Optimization 43, No. 2, 185–197 (1998; Zbl 0903.90162), A. M. Geoffrion, J. Math. Anal. Appl. 22, 618–630 (1968; Zbl 0181.22806), M. A. Hanson, J. Math. Anal. Appl. 80, 545–550 (1981; Zbl 0463.90080), Z. A. Khan and M. A. Hanson, J. Math. Anal. Appl. 205, No. 2, 330–336 (1997; Zbl 0872.90094), R. N. Mukherjee, J. Math. Anal. Appl. 162, No. 2, 309–316 (1991; Zbl 0751.90075), C. Singh and M. A. Hanson, J. Inf. Optimization Sci. 12, No. 1, 139–144 (1991; Zbl 0738.90065); C. Singh, S. K. Suneja and N. G. Rueda, J. Inf. Optimization Sci. 13, No. 2, 293–302 (1992; Zbl 0770.90059); S. K. Suneja and S. Gupta, Opsearch 27, No. 4, 239–253 (1990; Zbl 0719.90081); S. K. Suneja and M. K. Srivastava, Opsearch 31, No. 2, 127–143 (1994; Zbl 0815.90127); S. K. Suneja and C. S. Lalitha, Opsearch 30, No. 1, 1–14 (1993; Zbl 0793.90081); T. Weir, J. Inf. Optimization Sci. 7, 261–269 (1986; Zbl 0616.90080); Opsearch 25, No. 2, 98–104 (1988; Zbl 0655.90077); Util. Math. 36, 53–64 (1989; Zbl 0688.90052)]. In particular, [T. Antczak [J. Appl. Anal. 11, No. 1, 63–79 (2005; Zbl 1140.90485)] introduced the concept of differentiable V-\(r\)-invexity which is a generalization of invexity. He got the Kuhn-Tucker type necessary optimality theorem, weak, strong and strictly converse duality for a multiobjective optimization programming involving differentiable V-\(r\)-invex functions. The concepts of efficiency and proper efficiency play a key role in fractional vector optimization problems. Several authors including C. Singh and M. A. Hanson [J. Inf. Optimization Sci. 7, 41–48 (1986; Zbl 0593.90077)], N. Datta [J. Inf. Optimization Sci. 3, 262–268 (1982; Zbl 0498.90075)], R. N. Kaul and V. Lyall [Opsearch 26, No. 2, 108–121 (1989; Zbl 0676.90086)], J. C. Liu [Optimization 37, No. 1, 27–39 (1996; Zbl 0867.90095)] have discussed efficiency and proper efficiency to fractional vector minimization problems. In [J. Optimization Theory Appl. 53, 115–123 (1987; Zbl 0593.90071)] C. Singh derived the necessary conditions for efficient optimality of differentiable multiobjective programming under a constraint qualification.

MSC:

90C29 Multi-objective and goal programming
26A51 Convexity of real functions in one variable, generalizations
90C25 Convex programming
90C32 Fractional programming
90C46 Optimality conditions and duality in mathematical programming
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