Research On Two Types Of Interval Multi-Objective Optimization Method

In engineering fields,multi-objective optimization problems are common and often involve uncertain parameters represented by interval numbers,forming interval multiobjective optimization problems.In interval multi-objective optimization problems,the objective functions and constraint functions contain interval parameters,and it is impossible to directly obtain the interval range of uncertain functions by numerical calculation.For this problem,nested optimization or interval analysis are usually used to solve it.However,when dealing with complex interval multi-objective optimization problems with large intervals,multiple parameters,strong nonlinearity,etc.,the calculation accuracy and efficiency of interval analysis will be greatly reduced.Although nested optimization has better accuracy in the solving process,it also brings a considerable computational burden.In addition,when dealing with expensive black-box interval multi-objective optimization problems that can only be evaluated by simulation and require a lot of economic and time resources for a single evaluation,interval analysis and nested optimization are not applicable.For these two types of problems,this paper proposes two methods for interval multi-objective optimization.The main work and contributions of this paper are as follows:(1)For complex interval multi-objective optimization problems with large intervals,multiple parameters,strong nonlinearity,etc.,existing methods have defects of high computational cost and low solving accuracy.A fast interval multi-objective optimization method based on fuzzy rule base is proposed.This method uses the autonomous learning multi-model system to learn fuzzy rules and uses the obtained fuzzy rule base to represent the objective function and constraint possibility.The fuzzy rule base evaluation is used to replace the inner layer optimization to speed up the optimization process.This surrogate model processing method avoids time-consuming internal optimization and improves the applicability.At the same time,the method selects well-performing solutions from the candidate solution set generated from each generation of the fuzzy rule base and updates the model to improve its accuracy.In addition,the autonomous learning multi-model could process streaming data and can autonomously update fuzzy rule base.Therefore,there is no need to rebuild the model during the optimization iteration process,which has higher optimization efficiency than other surrogate models.Experiments on four benchmark test problems and one engineering case application show that the proposed algorithm has higher accuracy and applicability than interval analysis method,and higher efficiency and similar accuracy than nested optimization.(2)Considering that the existing interval multi-objective optimization methods cannot effectively deal with expensive black-box interval multi-objective optimization problems,this paper proposes a two-stage surrogate-assisted interval multi-objective optimization method.This method can effectively deal with such problems under the limitation of finite real evaluation times.Different from previous surrogate-assisted evolutionary algorithms,the proposed method explores solutions in two stages.The first stage uses a global model to replace the objective function and performs global exploration of the design space.The second stage establishes multiple local models for local exploration,thus avoiding the shortcomings of single exploration that may fall into local optimal solutions.In addition,in global exploration,a competitive swarm optimizer is used to optimize the model parameters to improve the model accuracy and thus improve the exploration ability of the algorithm.Finally,in the local exploration stage,the design space is divided into multiple subspaces.The solutions explored in the first stage are used to initialize the local exploration population and set ideal points.Then multiple local models are used to search subspaces and further exploit unexplored regions.The advantages of this method are verified by two sets of expensive black-box problem benchmark test sets commonly used by surrogate-assisted evolutionary algorithms.Finally,by limiting the real evaluation times of two interval multi-objective optimization problems,this method verifies its effectiveness in expensive black-box interval multi-objective optimization problems.