Genetic Algorithms For Two Classes Of Nonlinear Programming Problems With Interval Parameters

In the real-life world, there always exist some imprecise and uncertain elements which make the corresponding problems become uncertainty problems. When some uncertainty parameter is involved in an optimization problem, the problem is known as an uncertainty optimization problem. At present, uncertainty programming can be handled mainly in three manners: interval programming, stochastic programming or fuzzy numbers. In stochastic and fuzzy programming, it is sometimes difficult to specify an appropriate membership function or accurate probability distribution in an uncertain environment. Considering that both stochastic and fuzzy numbers can be transformed into intervals, interval programming become the most concerned one among these approaches. However, to date, most of research on interval programming is focused on the linear version of the problem, and there are few approaches to deal with nonlinear programs with interval parameters, especially to nonlinear cases involving bilevel optimization. In order to further investigate some efficient approaches for this kind of problems, in this thesis, we consider two classes of nonlinear programs with interval parameters and propose two efficient genetic algorithms for obtaining the optima.1. For a class of quadratic bilevel programming problems with interval objective coefficients in the leader’s and the lower’s levels, a genetic algorithm with two fitness functions is presented. Firstly, the coefficient interval of the lower-level objective is taken as the search space of the genetic algorithm. After doing so, for each individual,the lower level of the resulting problem doesn’t involve interval coefficients; In addition, the optimality conditions of quadratic programming are used to further transform the resulting problem into two certain quadratic programs. Furthermore,these two quadratic programs are solved by a base-enumerating method and its optimal values are taken as two fitness values. Finally, the best and the worst optimal solutions can be obtained by comparing two fitness values of all individuals,respectively. The simulation results show that the proposed algorithm is feasible and efficient.2. For nonlinear programming problems with interval parameters, a genetic algorithm based on a uniformly searching scheme is proposed. Firstly, the originalproblem is transformed into two exact bilevel programs. Secondly, the upper level variables are encoded as individuals, and these individuals are evaluated by solving the corresponding lower-level programs. Finally, in order to avoid producing similar offspring by inbreeding, a relative distance is adopted to provide a threshold value for crossover. Also, an orthogonal crossover operator with point oscillating is provided to generate offspring as uniformly as possible. The experimental results indicate that this algorithm is feasible and efficient.