A sufficient condition on the existence of a global weak sharp minima for general function in metric space is established. A characterization for convex function to have global weak sharp minima is also presented, whi...A sufficient condition on the existence of a global weak sharp minima for general function in metric space is established. A characterization for convex function to have global weak sharp minima is also presented, which generalized Burke and Ferris' result to infinite dimensional space. A characterization of the completeness of a metric space is given by the existence of global weak sharp minima.展开更多
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
Simulated annealing (SA) has been a very useful stochastic method for solving problems of multidimensional global optimization that ensures convergence to a global optimum. This paper proposes a variable cooling facto...Simulated annealing (SA) has been a very useful stochastic method for solving problems of multidimensional global optimization that ensures convergence to a global optimum. This paper proposes a variable cooling factor (VCF) model for simulated annealing schedule as a new cooling scheme to determine an optimal annealing algorithm called the Powell-simulated annealing (PSA) algorithm. The PSA algorithm is aimed at speeding up the annealing process and also finding the global minima of test functions of several variables without calculating their derivatives. It has been applied and compared with the SA algorithm and Nelder and Mead Simplex (NMS) methods on Rosenbrock valleys in 2 dimensions and multiminima functions in 3, 4 and 8 dimensions. The PSA algorithm proves to be more reliable and always able to find the optimum or a point very close to it with minimal number of iterations and computational time. The VCF compares favourably with the Lundy and Mees, linear, exponential and geometric cooling schemes based on their relative cooling rates. The PSA algorithm has also been programmed to run on android smartphone systems (ASS) that facilitates the computation of combinatorial optimization problems.展开更多
A novel gravity assist space pruning(GASP)algorithm based on image tools is proposed for solving interplanetary trajectory optimization problem.Compared with traditional GASP algorithm,the concept of image is introduc...A novel gravity assist space pruning(GASP)algorithm based on image tools is proposed for solving interplanetary trajectory optimization problem.Compared with traditional GASP algorithm,the concept of image is introduced to avoid missing interesting solutions with appropriate number of function evaluations.Image tools allow us to evaluate the objective function in regions in place of points and provide an effective way to evaluate the forward and backward constraints for the multi-gravity assist trajectory optimization problem.Since the interesting solutions of the interplanetary trajectory optimization problem are often clustered in a small portion of the search space rather than being overall evenly distributed,the regionwise evaluations with image tools make the little large interval with the proper Lipschitzian tolerances sampling effective.The detailed steps of the proposed method are presented and two examples including Earth Venus Mars(EVM)transfer and Earth Venus Venus Earth Jupiter Saturn(EVVEJS)transfer are given.Finally,a comparison with solutions given by the literature demonstrates the effectiveness of the proposed method.展开更多
基金The research was supported by the National Natural Science Foundation of China(10361008) Natural Science Foundation of Yunnan Province(2003A002M)
文摘A sufficient condition on the existence of a global weak sharp minima for general function in metric space is established. A characterization for convex function to have global weak sharp minima is also presented, which generalized Burke and Ferris' result to infinite dimensional space. A characterization of the completeness of a metric space is given by the existence of global weak sharp minima.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
文摘Simulated annealing (SA) has been a very useful stochastic method for solving problems of multidimensional global optimization that ensures convergence to a global optimum. This paper proposes a variable cooling factor (VCF) model for simulated annealing schedule as a new cooling scheme to determine an optimal annealing algorithm called the Powell-simulated annealing (PSA) algorithm. The PSA algorithm is aimed at speeding up the annealing process and also finding the global minima of test functions of several variables without calculating their derivatives. It has been applied and compared with the SA algorithm and Nelder and Mead Simplex (NMS) methods on Rosenbrock valleys in 2 dimensions and multiminima functions in 3, 4 and 8 dimensions. The PSA algorithm proves to be more reliable and always able to find the optimum or a point very close to it with minimal number of iterations and computational time. The VCF compares favourably with the Lundy and Mees, linear, exponential and geometric cooling schemes based on their relative cooling rates. The PSA algorithm has also been programmed to run on android smartphone systems (ASS) that facilitates the computation of combinatorial optimization problems.
基金supported by the National High Technology Research and Development Program (863)of China (2012AA121602)the National Natural Science Foundation of China(11078001)+1 种基金the Specialized Research Fund for the Doctoral Program of Higher Education of China (20133218120037)the Fundamental Research Funds for the Central Universities under Grant(NS2014091)
文摘A novel gravity assist space pruning(GASP)algorithm based on image tools is proposed for solving interplanetary trajectory optimization problem.Compared with traditional GASP algorithm,the concept of image is introduced to avoid missing interesting solutions with appropriate number of function evaluations.Image tools allow us to evaluate the objective function in regions in place of points and provide an effective way to evaluate the forward and backward constraints for the multi-gravity assist trajectory optimization problem.Since the interesting solutions of the interplanetary trajectory optimization problem are often clustered in a small portion of the search space rather than being overall evenly distributed,the regionwise evaluations with image tools make the little large interval with the proper Lipschitzian tolerances sampling effective.The detailed steps of the proposed method are presented and two examples including Earth Venus Mars(EVM)transfer and Earth Venus Venus Earth Jupiter Saturn(EVVEJS)transfer are given.Finally,a comparison with solutions given by the literature demonstrates the effectiveness of the proposed method.
