Non-differentiable multiobjective programming under generalised functions

摘要

In this paper, we consider a nonlinear multiobjective programming problem where functions involved are non-differentiable. The use of directional derivative in association with a hypothesis of an invex kind on the set has been of much interest in the recent past. Recently under preinvexity assumptions on f_i'(x_o; η_i(x, x_o)) and g_j'(x_o; θ_j(x, x_o)), necessary and sufficient optimality conditions for a non-smooth multiobjective optimisation problem under generalised class of d_I-V-type I functions have been derived. Working in this direction, we introduce a new class of (φ, d_I)-V-type I functions and illustrate through various non-trivial examples that this class is non-empty and extends some known classes introduced in the literature. Also, we obtain sufficient optimality conditions to enable a feasible solution of the primal problem to be its weak efficient/efficient solution. Then through an example, we illustrate the relevance of the sufficient optimality theorem obtained to get the efficient solution of the problem. Further, we formulate Wolfe type and Mond-Weir type multiobjective dual programs and establish various duality theorems under this newly introduced class of functions.