Duality for Some Nonlinear Fractional Programming Problems Under Generalized Convexity

Abstract

A common approach in direction of solving a mathematical programming problem is to transfer it from the primal domain to a dual domain. This method can prove to be very beneficial as the dual may have some simpler mathematical or geometrical structure. Besides, this transformation aids in getting the optimal solution with lesser algorithmic and computational efforts. As a dual provides a lower bound to the value of the primal problem, the second and higher-order duals give tighter bounds to the value of the primal problem. In this thesis higher-order dual models, for various forms of multiobjective fractional programming problems and a minimax fractional programming problem, are discussed. Convexity plays crucial role in the study of optimality conditions and duality theory in mathematical programming problems. With the rising complexity of mathematical problems, convexity is turning into a hard criteria to be achieved. Moreover not all the properties of convex functions are required to be satisfied in achieving optimality and duality results for mathematical programs. So we are interested in finding further general classes of functions which broaden the scope of optimality conditions and duality theorems in mathematical programming problems. In this direction, we introduce higher-order (C, α, γ, ρ, d) type-I functions, higher-order (C, α, γ, ρ, d) convex functions over cones, higher-order (Φ, ρ) convex functions over cones and higher-order (C, α, ρ, d) convex functionals. A multiobjective fractional programming problem containing support functions is considered. Then higher-order (C, α, γ, ρ, d) type-I functions are introduced which are inclusive of the various previously existing classes of generalized convex functions. For the considered problem a higher-order Schaible type dual is constructed and the validity of this dual model is attested with the help of duality theorems using the above mentioned class of generalized convexity. Further, an extension of the discussed work is made to the case of fractional programs in which ordering is defined with respect to cones. Another formulation of multiobjective fractional problems, in which the primal problem and the dual problem are symmetric to each other, are studied. Generally nonlinear programs are not symmetric. However assuming certain structural assumptions these programs can be modeled in the form of symmetric programs. On this line of action higher-order non-differentiable fractional symmetric programs over cones are studied. The relationship between the symmetric primal and dual problems is demonstrated with the help of generalized convex functions, namely higher-order K-(Φ, ρ) convex functions. A further discussed problem, involve the decision variables of the form of functions or curves, called as variational problem. This problem is a multiobjective fractional variational symmetric problem, in which higher-order functional approximations are used. Two type of generalized convex functions named as higher-order (F, α, ρ, d) \(C, α, ρ, d) convex functionals are introduced. Then the duality results are obtained for the symmetric programs with the help of generalized convex functionals. In minimax programming problems, we consider a non-differentiable fractional minimax program containing support functions. For this problem higher-order parametric dual model is structured and by means of higher-order (C, α, γ, ρ, d) convexity we attain duality relations between the minimax fractional problem and the parametric higher-order dual problems. Then we study differentiable multiobjective programming problems in which the constraints are affected by data uncertainty. We assume that the uncertainty parameters lie in compact convex sets. These problems are solved using a deterministic technique for dealing with uncertainties, called robust optimization approach. So the problem with data uncertainty is transformed into a robust optimization problem which considers all the elements from uncertainty set including the worst possible case. Then for this robust problem we study sufficient optimality conditions for robust efficient solutions which are placid even under data uncertainty. Finally a robust dual model is constructed for the robust problem and the duality theorems are proved using pseudoconvexity and quasiconvexity assumptions. These results are further extended for multiobjective fractional programs under data uncertainty.