DUALITY IN NONLINEAR VECTOR OPTIMIZATION
LOH WING WAH
LOH WING WAH
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Abstract
In this thesis, we attempted to give a systematic account of duality theory for a convex vector optimization nonlinear programming problem, In mathematical programming, duality means in association to, say a minimization problem, we maximize a related problem, called its dual. Occasionally, the dual problem may turn out to be an easier problem to solve and the solution to the original problem can be obtained via the solution of its dual.
In Chapter I, we discussed some elementary concepts of vector optimization and duality theory. Standard theorems of alternatives, separation hyperplane and nonlinear duality were also given.
In Chapter II, we presented a duality theory using the vector space approach. Under suitable assumptions, the weak, strong and strict converse duality theorems were proved. The specialization of the theory to linear multiple objective programming generalised the well-known dual problem of Isermann.
In Chapter III, a duality principle derived from abstract optimization is developed in a general setting. In this formulation, weak duality was assumed by way of definition of the dual set. Strong duality theorem followed easily and with some additional assumptions, a strict converse duality theorem was proved. With the aid of certain continuous linear functionals, the dual set was transformed in such a way that connection with known duality results in multi-objective linear programming could be demonstrated.
In Chapter IV, the duality theory associated to efficient points a matrix, rather than a vector, of dual variables. In relation to the Linear Approximation Problem, a duality result of Isermann for a linear multiple objective optimization problem was discussed. Weak and strong duality theorems were proved. We also discussed Kuhn-Tucker condition and connection to the parametric approach was made.
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1991
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Thesis