Jiu Ding Bingsheng He Qin Ni Wenyu Sun
Numerical Algebra, Control & Optimization 2013, 3(2): i-ii doi: 10.3934/naco.2013.3.2i
This special issue is dedicated to the memory of the late Professor Xuchu He of Nanjing University, China. Professor He passed away on April 30, 1990 at the age of 69. An international workshop was held at Nanjing Normal University in May 2011 on the occasion of his 90th birthday.

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A quasi-Newton trust region method based on a new fractional model
Honglan Zhu Qin Ni Meilan Zeng
Numerical Algebra, Control & Optimization 2015, 5(3): 237-249 doi: 10.3934/naco.2015.5.237
In this paper, a general fractional model is proposed. Based on the fractional model, a quasi-Newton trust region algorithm is presented for unconstrained optimization. The trust region subproblem is solved in the simplified way. We discussed the choices of the parameters in the fractional model and prove the global convergence of the proposed algorithm. Some primary test results shows the feasibility and validity of the fractional model.
keywords: fractional model global convergence. Unconstrained optimization quasi-Newton method trust region method
A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems
Guanghui Zhou Qin Ni Meilan Zeng
Journal of Industrial & Management Optimization 2017, 13(2): 595-608 doi: 10.3934/jimo.2016034

In this paper, a scaled method that combines the conjugate gradient with moving asymptotes is presented for solving the large-scaled nonlinear unconstrained optimization problem. A diagonal matrix is obtained by the moving asymptote technique, and a scaled gradient is determined by multiplying the gradient with the diagonal matrix. The search direction is either a scaled conjugate gradient direction or a negative scaled gradient direction under different conditions. This direction is sufficient descent if the step size satisfies the strong Wolfe condition. A global convergence analysis of this method is also provided. The numerical results show that the scaled method is efficient for solving some large-scaled nonlinear problems.

keywords: Scaling conjugate gradient moving asymptotes the Wolfe condition unconstrained optimization
Subspace trust-region algorithm with conic model for unconstrained optimization
Xin Zhang Jie Wen Qin Ni
Numerical Algebra, Control & Optimization 2013, 3(2): 223-234 doi: 10.3934/naco.2013.3.223
In this paper, a new subspace algorithm is proposed for unconstrained optimization. In this new algorithm, the subspace technique is used in the trust region subproblem with conic model, and the dogleg method is modified to solve this subproblem. The global convergence of this algorithm under some reasonable conditions is established. Numerical experiment shows that this algorithm may be superior to the corresponding algorithm without using subspace technique especially for large scale problems.
keywords: conic model trust region method Unconstrained optimization subspace method global convergence.

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