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Research On Intelligent Algorithms For Constrained Interval Nonlinear Optimizations

Posted on:2016-11-01Degree:DoctorType:Dissertation
Country:ChinaCandidate:Y ZhaoFull Text:PDF
GTID:1108330482454729Subject:Instrument Science and Technology
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Widely uncertainty exists in the practical detection engineering problems,and researching on the theories and algorithms of uncertain optimization is very significant for the reliability design of industrial products and systems.Interval numbers optimization is a kind of relatively newly, developed uncertain optimization method,in which interval is used to model the uncertainty of a variable.Thus the variation bounds of the uncertain variables are only needed,which can be obtained through a little amount of uncertainty information. Interval multi-objective optimization in engineering practice and scientific research is an important issue, and has a high value. Evolutionary algorithm, a kind of swarm intelligence method is very suitable for multi-objective optimization problem, so evolutionary algorithm for multi-objective optimization has become a hot topic in evolutionary computing. Many famous researchers focus the interval number optimization, but some key technical difficulties remain in nonlinear interval number optimization. So this dissertation conducts a systematical research for the interval number optimization.As a result,the following studies are finished in this dissertation:1.The dissertation describes the concept of interval number, interval number comparison method and interval number four operations, etc. Then it proposes a kind of distance formula of interval value for interval numbers and proves the nature of its positive definiteness, symmetry and triangle inequality and so on.2.A kind of algorithm for constrained single objective nonlinear optimization problems with interval numbers is proposed. Firstly, linear approximation models with respect to the uncertain variables are made for the uncertain objective function and constraints using the first-order Taylor expansion,and a linear interval optimization problem is got. Secondly, by weighting method the two objective functions in interval optimization become deterministic single objective function, then the constrained optimization will be converted into unconstrained optimization by penalty function method. Finally, genetic algorithm is used to solve unconstrained optimization problems improved. Based on the uniform framework of swarm intelligence computation, swarm intelligence computation mode for genetic algorithm is given, so it is easy to analyze and modify the algorithm. The effect of uncertainty is given in simulation.3.Based on interval approximation conversion and penalty function,a kind of algorithm for the constrained multi-objective nonlinear optimization problems with interval numbers is proposed. Firstly, linear approximation models with respect to the uncertain variables are made for the uncertain objective function and constraints using the first-order Taylor expansion,and a linear interval number optimization problem is obtained. Secondly, by weighting method each objective function of interval value in interval multi-objective optimization become deterministic single objective function, then the constrained optimization will be converted into unconstrained optimization by penalty function method. Finally, NSGA-II is used to solve unconstrained optimization problems improved. Based on the uniform framework of swarm intelligence computation, swarm intelligence computation mode for NSGA-II is given, so it is easy to analyze and modify the algorithm. The stability of the proposed algorithm is proved by Markov stochastic process theory. The important parameters of the algorithm are analyzed in simulation.and it is presented how the different important parameters impact on the optimization final value in simulation.4.Based on interval possibility degree and interval distance, NSGA-II for multi-objective optimization problems with interval numbers is proposed from the deterministic NSGA-II. Firstly, linear approximation models with respect to the uncertain variables are made for the uncertain objective function and constraints using the first-order Taylor expansion,and a linear interval number optimization problem is got. P dominance relationship is defined based on the interval possibility degree, which is applied in getting the rank values of solutions; the proposed algorithm utilizes the interval distance formula, instead of the crowding distance formula, to evaluate the interval crowding distance of solutions of the same rank value then sorts solutions in order of their interval crowding distance. In order to increase the search area of the population, this algorithm proposes a crossover operator based on the normal distribution. A constrained elite strategy, in which the constraint violation degree of individual is compared with the allowable constraint violation degree, is used to select the solutions of correspondingly satisfying constraint from the population. The stability of the proposed algorithm is proved by Markov stochastic process theory. The important parameters of the algorithm are analyzed in simulation.and it is presented how The different important parameters impact on the optimization final value in simulation.5.In order to get the satisfactory solutions in the pareto front that conforms to the decision-makers preference, a kind of multiple attributes decision making interval TOPSIS is presented based on NSGA-II. The multi-attribute decision-making model is made, which regards the selected individuals as the alternatives set, the objective function as a set of attributes, and the preference of every objective function as attribute weights. Then distance formula for interval numbers in the dissertation is used in interval TOPSIS,so the satisfactory solutions in pareto front which conform to the decision-makers preference are got.Innovative work in this paper is as follows.1. The distance formula of interval value is proposed and the nature of its positive definiteness, symmetry and triangle inequality are proved. Compared with the deterministic distance formula in other paper, distance formula of interval value contains more uncertain information.2. Based on interval approximation conversion and penalty function,a kind of algorithm for constrained nonlinear optimization problems with interval numbers is proposed. Firstly,linear approximation models with respect to the uncertain variables are made for the uncertain objective function and constraints using the first-order Taylor expansion,and a linear interval optimization problem is got. Secondly, by weighting method each objective function in interval multi-objective optimization become deterministic single objective function, then the constrained optimization will be converted into unconstrained optimization by penalty function method. Finally, genetic algorithm and NSGA-II are used to solve unconstrained single objective optimization problems and multi-objective optimization problems transformed.3. NSGA-II for multi-objective optimization problems with interval numbers is proposed from the deterministic NSGA-II. P dominance relationship is defined based on the interval possibility degree, which is applied in getting the rank values of solutions; the proposed algorithm utilizes the interval distance formula, instead of the crowding distance formula, to evaluate the interval crowding distance of solutions of the same rank value then sorts solutions in order of their interval crowding distance. In order to increase the search area of the population, this algorithm proposes a crossover operator based on the normal distribution. A constrained elite strategy, in which the constraint violation degree of individual is compared with the allowable constraint violation degree, is used to select the solutions of correspondingly satisfying constraint from the population.4. In order to obtain satisfactory solutions in pareto front that conforms to the decision-makers preference, a kind of multiple attributes decision making interval TOPSIS is presented. A multi-attribute decision-making model is made, which regards the selected individuals as the alternatives set, the objective function as a set of attributes, and the preference of every objective function as attribute weights. Then distance formula for interval numbers in the dissertation is used in interval TOPSIS,so the satisfactory solutions in pareto front which conform to the decision-makers preference are got.
Keywords/Search Tags:Interval programming, Multi-objective optimization, Genetic algorithms, NSGA-II, Multiple attribute decision making, TOPSIS
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