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, and#945;, and#947;, and#961;, d) type-I functions, higher-order (C, and#945;, and#947;, and#961;, d) convex functions over cones, higher-order (and#934;, and#961;) convex functions over cones and higher-order (C, and#945;, and#961;, d) convex functionals. A multiobjective fractional programming problem containing support functions is considered. Then higher-order (C, and#945;, and#947;, and#961;, 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.