This paper proposes the use of the flexible tolerance method(FTM) modified with scaling of variables and hybridized with different unconstrained optimization methods to solve real constrained optimization problems.The...This paper proposes the use of the flexible tolerance method(FTM) modified with scaling of variables and hybridized with different unconstrained optimization methods to solve real constrained optimization problems.The benchmark problems used to analyze the performance of the methods were taken from G-Suite functions.The original method(FTM) and other four proposed methods:(i) FTM with scaling of variables(FTMS),(ii) FTMS hybridized with BFGS(FTMS-BFGS),(iii) FTMS hybridized with modified Powell's method(FTMS-Powell)and(iv) FTMS hybridized with PSO(FTMS-PSO), were implemented. The success rates of the methods were 80%,100%, 75%, 95% and 85%, for FTM, FTMS, FTMS-BFGS, FTMS-Powell and FTMS-PSO, respectively. Numerical experiments including real constrained problems indicated that FTMS gave the best performance, followed by FTMSPowell and FTMS-PSO. Despite the inferior performance compared to FTMS and FTMS-Powell, the FTMS-PSO method presented some advantages since good different initial points could be obtained, which allow exploring different routes through the solution space and to escape from local optima. The proposed methods proved to be an effective way of improving the performance of the original FTM.展开更多
The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessi...The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex （or even uniformly convex）. We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.展开更多
In this paper, a switching method for unconstrained minimization is proposed. The method is based on the modified BFGS method and the modified SR1 method. The eigenvalues and condition numbers of both the modified upd...In this paper, a switching method for unconstrained minimization is proposed. The method is based on the modified BFGS method and the modified SR1 method. The eigenvalues and condition numbers of both the modified updates are evaluated and used in the switching rule. When the condition number of the modified SR1 update is superior to the modified BFGS update, the step in the proposed quasi-Newton method is the modified SR1 step. Otherwise the step is the modified BFGS step. The efficiency of the proposed method is tested by numerical experiments on small, medium and large scale optimization. The numerical results are reported and analyzed to show the superiority of the proposed method.展开更多
Global Positioning System (GPS) meteorology data variational assimilation can be reduced to the problem of a large-scale unconstrained optimization. Because the dimension of this problem is too large, most optimal alg...Global Positioning System (GPS) meteorology data variational assimilation can be reduced to the problem of a large-scale unconstrained optimization. Because the dimension of this problem is too large, most optimal algorithms cannot be performed. In order to make GPS/MET data assimilation able to satisfy the demand of numerical weather prediction, finding an algorithm with a great convergence rate of iteration will be the most important thing. A new method is presented that dynamically combines the limited memory BFGS (L-BFGS) method with the Hessian-free Newton(HFN) method, and it has a good rate of convergence in iteration. The numerical tests indicate that the computational efficiency of the method is better than the L-BFGS and HFN methods.展开更多
基金CAPES(Coordenacao de Aperfeicoamento de Pessoal de Nível Superior)CNPq(Conselho Nacional de Desenvolvimento Científicoe Tecnológico,grant number 161464/2013-0)for the financial support
文摘This paper proposes the use of the flexible tolerance method(FTM) modified with scaling of variables and hybridized with different unconstrained optimization methods to solve real constrained optimization problems.The benchmark problems used to analyze the performance of the methods were taken from G-Suite functions.The original method(FTM) and other four proposed methods:(i) FTM with scaling of variables(FTMS),(ii) FTMS hybridized with BFGS(FTMS-BFGS),(iii) FTMS hybridized with modified Powell's method(FTMS-Powell)and(iv) FTMS hybridized with PSO(FTMS-PSO), were implemented. The success rates of the methods were 80%,100%, 75%, 95% and 85%, for FTM, FTMS, FTMS-BFGS, FTMS-Powell and FTMS-PSO, respectively. Numerical experiments including real constrained problems indicated that FTMS gave the best performance, followed by FTMSPowell and FTMS-PSO. Despite the inferior performance compared to FTMS and FTMS-Powell, the FTMS-PSO method presented some advantages since good different initial points could be obtained, which allow exploring different routes through the solution space and to escape from local optima. The proposed methods proved to be an effective way of improving the performance of the original FTM.
基金supported by NSFC 10001031 and 70472074supported by NSERC Grant 283103
文摘The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex （or even uniformly convex）. We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.
基金This work is supported by National Natural Key product Foundations of China 10231060.
文摘In this paper, a switching method for unconstrained minimization is proposed. The method is based on the modified BFGS method and the modified SR1 method. The eigenvalues and condition numbers of both the modified updates are evaluated and used in the switching rule. When the condition number of the modified SR1 update is superior to the modified BFGS update, the step in the proposed quasi-Newton method is the modified SR1 step. Otherwise the step is the modified BFGS step. The efficiency of the proposed method is tested by numerical experiments on small, medium and large scale optimization. The numerical results are reported and analyzed to show the superiority of the proposed method.
基金the National Excellent Youth Fund(Grant No.49825109)the CAS Key Innovation Direction Project(Grant No.KZCX2-208),and LASG Project.
文摘Global Positioning System (GPS) meteorology data variational assimilation can be reduced to the problem of a large-scale unconstrained optimization. Because the dimension of this problem is too large, most optimal algorithms cannot be performed. In order to make GPS/MET data assimilation able to satisfy the demand of numerical weather prediction, finding an algorithm with a great convergence rate of iteration will be the most important thing. A new method is presented that dynamically combines the limited memory BFGS (L-BFGS) method with the Hessian-free Newton(HFN) method, and it has a good rate of convergence in iteration. The numerical tests indicate that the computational efficiency of the method is better than the L-BFGS and HFN methods.
