%0 Journal Article
%T Multiobjective Optimization Involving Quadratic Functions
%A Oscar Brito Augusto
%A Fouad Bennis
%A Stephane Caro
%J Journal of Optimization
%D 2014
%R 10.1155/2014/406092
%X Multiobjective optimization is nowadays a word of order in engineering projects. Although the idea involved is simple, the implementation of any procedure to solve a general problem is not an easy task. Evolutionary algorithms are widespread as a satisfactory technique to find a candidate set for the solution. Usually they supply a discrete picture of the Pareto front even if this front is continuous. In this paper we propose three methods for solving unconstrained multiobjective optimization problems involving quadratic functions. In the first, for biobjective optimization defined in the bidimensional space, a continuous Pareto set is found analytically. In the second, applicable to multiobjective optimization, a condition test is proposed to check if a point in the decision space is Pareto optimum or not and, in the third, with functions defined in n-dimensional space, a direct noniterative algorithm is proposed to find the Pareto set. Simple problems highlight the suitability of the proposed methods. 1. Introduction Life is about making decisions and the choice of the optimal solutions is not an exclusive subject for scientists, engineers, and economists. Decision making is present in day-to-day life. Looking for an enjoyable vacancy, everyone will formulate an optimization problem to a travel agent, a problem like with a minimum amount of money visit a maximum number of places in a minimum amount of time and with the maximum level of comfort. Usually all real design problems have more than one objective; namely, they are multiobjective. Moreover, the design objectives are often antagonistic. Edgeworth [1] was the pioneer to define an optimum for multicriteria economic decision making problem, at King¡¯s College, London. It was about the multiutility problem within the context of two consumers, and . ¡°It is required to find a point such that in whatever direction we take an infinitely small step, and do not increase together but that, while one increases, the other decreases.¡± Few years later, in 1896, Pareto [2], at the University of Lausanne, Switzerland, formulated his two main theories, Circulation of the Elites and the Pareto optimum. ¡°The optimum allocation of the resources of a society is not attained so long as it is possible to make at least one individual better off in his own estimation while keeping others as well off as before in their own estimation.¡± Since then, many researchers have been dedicated to developing methods to solve this kind of problem. Interestingly, solutions for problems with multiple objectives, also called
%U http://www.hindawi.com/journals/jopti/2014/406092/