Duality in Multiobjective Programming
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Date
2012-01
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Addis Ababa University
Abstract
Nowadays, human beings have been confronted with multiple criteria decision making problems. We want to have a good life, which may mean more wealth, more power, more respect and more time for our selves, together with a good health and a good second generation, e.t.c. Unlike single objective optimization in solving multi-objective optimization problem, we have solution set that is called efficient set. It is from this set decision is made by taking elements of efficient set as alternatives, which is given by analysts. This graduate project report contains the mathematical theories in multi-objective optimization, necessary and sufficient condition for existence of efficient solutions and their properties in partial ordered vector space. Furthermore, the dual problem has (under additional conditions) the same optimal value as the given “primal” optimization problem, but solving the dual problem could be done with other methods of analysis or numerical mathematics. An approximate solution of the given minimization problem gives an estimation of the minimal value ����∗ from above, whereas an approximate solution of the dual problem is an estimation of ����∗ from below, so that one gets intervals which contain ����∗ . Lagrange method, saddle points, equilibrium points of two person games, shadow prices in economics, perturbation methods or dual variational principles, it becomes clear that optimal dual variables often have a special meaning for the given problem. Thus, in this report different duality approach for multiobjective optimization problem is discussed using point-to-set map.
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Duality in Multiobjective Programming