Article: Non-differentiable multiobjective programming under generalised functions Journal: International Journal of Operational Research (IJOR) 2015 Vol.23 No.3 pp.363 - 385 Abstract: 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 fi′(xo;ηi(x,xo)) and g′j (xo;θj(x,xo)), necessary and sufficient optimality conditions for a non-smooth multiobjective optimisation problem under generalised class of dI-V-type I functions have been derived. Working in this direction, we introduce a new class of (φ, dI)-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. Inderscience Publishers - linking academia, business and industry through research

Title: Non-differentiable multiobjective programming under generalised functions

Authors: Pallavi Kharbanda; Divya Agarwal; Deepa Sinha

Addresses: Centre for Mathematical Sciences, Banasthali University, Banasthali, Rajasthan – 304022, India ' Department of Applied Sciences and Humanities, ITM University, Sector 23A, Gurgaon, Haryana – 122017, India ' Department of Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi – 110021, India

Abstract: 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.

Keywords: multiple-objective programming; d_I-V-type I functions; directional derivative; Lipschitz function.

DOI: 10.1504/IJOR.2015.069627

International Journal of Operational Research, 2015 Vol.23 No.3, pp.363 - 385

Received: 09 Jan 2013
Accepted: 07 Aug 2013
Published online: 29 May 2015 *

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Title: Non-differentiable multiobjective programming under generalised functions

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