Multi-Objective Optimization Based On Adaptation Learning Mechanism Of Covariance Matrix

The research of multi-objective optimization problem is mainly to improve the non-dominated solution set to close the Pareto optimal solution,but the algorithm applied to multi-objective optimization has problems such as premature convergence and poor acquisition efficiency of non-dominated solution set,especially the influence of rotating space on the target non-dominated set.For this problem,this paper applies the covariance matrix adaptation learning mechanism(CMA-ES variant,that is elite CMA-ES)to the multi-objective optimization problem,and solves the problem of premature convergence and poor efficiency of non-dominated solution set for multiobjective optimization problems.The work done is as follows:(1)Aiming at the shortcomings of Cholesky factor update standard CMA-ES,the Cholesky factor rank-? update covariance matrix method is proposed.The Cholesky factor rank-? update method was combined with the Cholesky factor rank-1 update method using standard CMA-ES and Active-CMAES and compared to other CMA-ES variants.The experimental results show that the proposed Cholesky factor rank-? update method effectively improves the rank-? update time of the covariance matrix.After applying the standard CMA-ES and Active-CMAES,the running time of the algorithm is faster than the original algorithm.(2)A new cumulative evolutionary path is proposed for the deficiency of triangular Cholesky factor rank-1 update CMA-ES.The new cumulative evolutionary path replaces the conjugate evolutionary path update step size when the Cholesky factor is updated by the triangular Cholesky factor,and it is no longer necessary to calculate the inverse triangular Cholesky factor.To verify the efficiency of the improved triangular Cholesky factor rank-1 update CMA-ES,the time and efficiency of updating CMA-ES with both Cholesky factor rank-1 and rank-? were compared.The experiment selects the benchmark function to verify the simulation.The experimental results show that the new cumulative evolution path combined with the triangular Cholesky factor update covariance matrix,the step value is not affected,and the running time of the algorithm is faster than the original algorithm,and the objective function value remains optimal.(3)Based on the work of(1)and(2),the elite CMA-ES algorithm is improved.One is to use the auxiliary evolutionary path to improve the time complexity of the Cholesky factor rank-1 update in the elite CMA-ES.The second is to apply the Cholesky factor rank-? update model to the elite CMA-ES,and compare the efficiency of the improved elite CMA-ES with other elite CMA-ES variants.The experimental results show that the improved elite CMA-ES has better performance.(4)The improved elite CMA-ES algorithm in(3)is combined with nondominated sorting and crowding distance to solve the multi-objective optimization problem to form the MOCholCMA algorithm,and the MOCholCMA algorithm and other classical multi-objective evolution algorithms are compared in the multi-objective test function set.The experimental results show that the MOCholCMA algorithm has good convergence,IGD index and DM index.The Pareto Front of the non-dominated solution set is close to the real Pareto Front.The MOCholCMA algorithm inherits the invariant properties of the CMA-ES algorithm and is invariant to the rotation space.