Generalized Invexity of Higher Order and Its Applications in Variational Problems

In the present paper the definition of invexity for continuous functions is extended to invexity of order m which is further generalized to ρ-pseudoinvexity type I of order m, ρ-pseudoinvexity type II of order m, as well as ρ-quasi invexity type I of order m and ρ-quasiinvexity type II of order m. The central objective of the paper is to study variational problem where the functionals involved satisfy the above stated generalized ρ-invexity assumptions of order m. Wolfe type and Mond Weir type of duals are formulated. Sufficient optimality conditions and duality results are proved. It is demonstrated with the help of an example that the class of ρ-invex functionals of order m is more general than the class of ρ-invex functionals. Further, it may be noted that the results presented in this paper are more powerful than the existing results as this new class of functions defined here satisfies mth derivative test.