We make some ecient modications on the modied secant equation proposed by Zhangand Xu (2001). Then we introduce modied BFGS method using propose secant equation, and obtain some attractive results in theory and practice. We establish the global con-vergence property of the proposed method without convexity assumption on the objectivefunction. Numerical results on some testing problems from CUTEr collection show the pri-ority of the proposed method to some existing modied secant methods in practice.

Following the setting of the Dai-Liao (DL) parameter in conjugate gradient (CG) methods, we introduce two new parameters based on the Modified secant equation proposed by Li et al. (Comput. Optim. Appl. 202: 523-539, 2007) with two approaches, which use an extended new conjugacy Condition. The first is based on a Modified descent three-term search direction, as the descent Hestenes-Stiefel CG method. The second is based on the quasi-Newton (QN) approach. Global convergence of the proposed methods for uniformly convex functions and general functions is proved. Numerical experiments are done on a set of test functions of the CUTEr collection and the results are compared with some well-known methods.

Using Taylor's series we propose a Modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this Modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and func-tion values while the usual secant relation uses only gradient values. Under appropriate Conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-More performance profiles.

In this paper, an efficient conjugate gradient method for unconstrained optimization is introduced. Parameters of the method are obtained by solving an optimization problem, and using a variant of the Modified secant Condition. The new conjugate gradient parameter benefits from function information as well as gradient information in each iteration. The proposed method has global convergence under mild assumptions. Using a collection of CUTEr problems, the method is compared with some existing algorithms to show its effectiveness.

In this article the general non-symmetric parametric form of the incremental secant stiffness matrix for nonlinear analysis of solids have been investigated to present a semi analytical sensitivity analysis approach for geometric nonlinear shape optimization. To approach this aim the analytical formulas of secant stiffness matrix are presented. The models were validated and used to perform investigating different parameters affecting the shape optimization. Numerical examples utilized for this investigating sensitivity analysis with detailed discussions presented.

With respect to importance of the conjugate gradient methods for large-scale optimization, in this study a descent three-term conjugate gradient method based on an extended Modified secant Condition is proposed. In the proposed method, objective function values are used in addition to the gradient information. Also, it is established that the method is globally convergent without convexity assumption on the objective function. Numerical experiments show reasonable computational efficiency of the suggested method.

Monitoring and evaluation of transformers are essential to prevent their insulation failure. In this paper, the use of 2-furan-carboxaldehyde (2-FAL) and methanol (MeOH) concentrations as the main products of paper insulation degradation, and insulation Condition markers, has been studied. In order to study the degradation process and production of degradation products, the thermal aging process of the transformer insulation system was implemented in laboratory Conditions. The results of laboratory studies show that in the early stages of degradation the amount of MeOH is significant compared to 2-FAL. Also, the estimation of the degree of polymerization (DP) in the early stages of degradation (DP>800) through MeOH concentration and with decreasing DP (DP <800) through 2-FAL concentration is closer to the real value. The results of studies performed on 35 distribution transformers confirm the production of significant amounts of MeOH in the early stages of degradation. Also, the estimated DP values for the studied transformers were obtained through 2-FAL and MeOH concentration. The results show that estimating the amount of DP through MeOH concentration is associated with a probability of error of about 9% compared to estimating DP through 2-FAL concentration.