A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided differenc...A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.展开更多
To predict the Elasto-Plastic Behaviors(EPBs)of aligned inclusions reinforced composites,this paper develops an interpolative Mori-Tanaka/Double-Inclusion(MT-DI)homogenization model with the secant formulation,and giv...To predict the Elasto-Plastic Behaviors(EPBs)of aligned inclusions reinforced composites,this paper develops an interpolative Mori-Tanaka/Double-Inclusion(MT-DI)homogenization model with the secant formulation,and gives the numerical implementation algorithms of the developed MT-DI model with the secant formulation.The Finite Element(FE)homogenization method is implemented to provide the "exact" EPBs of the composites and thus validate the MT-DI model with the secant formulation.The MT-DI model with the 2 nd-order secant formulation is validated to provide the more accurate predictions,while the MT-DI model with the 1 st-order secant formulation always gives the stiffer predictions.The results show that using the macro-stress and macrostrain as the inputs,the MT-DI model with the secant formulation gives the identical predictions.The predictions of the MT-DI model with the secant formulation vary between those of the MT and DI models with the secant formulation.Meanwhile,the MT-DI model with the secant formulation does not predict the accurate EPBs for the phases of the composites.展开更多
Deflection is a significant indicator of bridge’s strength and its whole stiffness, so the research on deflection measurement is an important aspect of bridge health monitoring. There have existed many measurement me...Deflection is a significant indicator of bridge’s strength and its whole stiffness, so the research on deflection measurement is an important aspect of bridge health monitoring. There have existed many measurement methods of bridge deflection so far, while inclination method is gradually catching more and more attention for its fair obviously comprehensive advantages. However, the inclination method at present focuses on measuring the rotation of bridge’s section at testing point, that is, the tangent angle of deflection curve. With the tangent angle, the deflection curve can be determined by the methods of curve fitting or (and) integration or conjugate beam. The methods mentioned above, are not only complicated in calculation, but also bad in accuracy. The deflection measurement method proposed by this paper is based on measuring the inclination of two points initiatively in horizontal line, that is, the secant angle of the deflection curve, and on the simple triangle function operation. The proposed method is simple in theory, but good in accuracy for either static or dynamic load. The numerical simulation suggests that the error of the proposed method is less than 1%.展开更多
In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximat...In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.展开更多
The secant methods discussed by Fontecilla (in 1988) are considerably revised through employing a trust region multiplier strategy and introducing a nondifferentiable merit function. In this paper the secant methods a...The secant methods discussed by Fontecilla (in 1988) are considerably revised through employing a trust region multiplier strategy and introducing a nondifferentiable merit function. In this paper the secant methods are also improved by adding a dogleg typed movement which allows to overcome a phenomena similar to the Maratos effect. Furthermore, these algorithms are analyzed and global convergence theorems as well as local superlinear convergence rate are proved.展开更多
Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we ...Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we also provide an error estimate that matches the convergence order of the two-step secant method.At last,we give an application of the proposed theorem.展开更多
A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis co...A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x^＊, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.展开更多
In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmo...In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.展开更多
In this paper, we present m time secant like multi projection algorithm for sparse unconstrained minimization problem. We prove this method are all q superlinearly convergent to the solution about m≥1 . At last, we f...In this paper, we present m time secant like multi projection algorithm for sparse unconstrained minimization problem. We prove this method are all q superlinearly convergent to the solution about m≥1 . At last, we from some numerical results, discuss how to choose the number m to determine the approximating matrix properly in practical use.展开更多
In this paper, we discuss the relationship between the sparse symmetric Broyden (SPSB) method [1, 2] and m-time secant-like multi-projection (SMP) method [3] and prove that when m goes to infinity, the SMP method is c...In this paper, we discuss the relationship between the sparse symmetric Broyden (SPSB) method [1, 2] and m-time secant-like multi-projection (SMP) method [3] and prove that when m goes to infinity, the SMP method is corresponding to the SPSB method.展开更多
By making a convex combination of the modified secant equations proposed by Yuan and Wei et al.,a hybrid secant equation and also,a modified BFGS algorithm is proposed.The hybridization parameter is effectively comput...By making a convex combination of the modified secant equations proposed by Yuan and Wei et al.,a hybrid secant equation and also,a modified BFGS algorithm is proposed.The hybridization parameter is effectively computed using the available information of recent iterations.Under proper conditions,it is shown that the proposed algorithm is globally,locally and superlinearly convergent.By using the performance profile introduced by Dolan and Mor6,a comparison between the implementations of the proposed algorithm and two efficient modified BFGS algorithms proposed by Yuan and Wei et al.,on a set of unconstrained optimization test problems from the CUTEr collection,is done.Numerical results demonstrating the efficiency of the proposed modified BFGS algorithm are reported.展开更多
This paper formulates and analyzes a line search method for general nonlinear equalityconstrained optimization based on filter methods for step acceptance and secant methods for searchdirection.The feature of the new ...This paper formulates and analyzes a line search method for general nonlinear equalityconstrained optimization based on filter methods for step acceptance and secant methods for searchdirection.The feature of the new algorithm is that the secant algorithm is used to produce a searchdirection,a backtracking line search procedure is used to generate step size,some filtered rules areused to determine step acceptance,second order correction technique is used to reduce infeasibility andovercome the Maratos effect.Global convergence properties of this method are analyzed:under mildassumptions it is showed that every limit point of the sequence of iterates generated by the algorithmis feasible,and that there exists at least one limit point that is a stationary point for the problem.Moreover,it is also established that the Maratos effect can be overcome in our new approach by addingsecond order correction steps so that fast local superlinear convergence to a second order sufficient localsolution is achieved.Finally,the results of numerical experiments are reported to show the effectivenessof the line search filter secant method.展开更多
The failure to achieve minimum design overlap between secant piles compromises the ability of a structure to perform as designed,resulting in water leakage or even ground collapse.To establish a more realistic simulat...The failure to achieve minimum design overlap between secant piles compromises the ability of a structure to perform as designed,resulting in water leakage or even ground collapse.To establish a more realistic simulation and provide guidelines for designing a safe and cost-effective secant-pile wall,a three-dimensional model of a secant pile,considering the geometric imperfections of the diameter and direction of the borehole,is introduced.An ultrasonic cross-hole test was performed during the construction of secant piles in a launching shaft in Beijing,China.Based on the test results,the statistical characteristics of the pile diameters and orientation parameters were obtained.By taking the pile diameter D,inclination angleβ,and azimuth angleαas random variables,Monte Carlo simulations were performed to discuss the influence of different design parameters on the probability density functions of the overlap of secant piles.The obtained results show that the randomness of the inclination angle and pile diameter can be well described by a normal distribution,whereas the azimuth angle is more consistent with a uniform distribution.The integrity of the secant-pile wall can be overestimated without considering the uncertainty of geometric imperfections.The failure of the secant-pile wall increases substantially with increasing spatial variability in drilling inclination and diameter.A design flowchart for pile spacing under the target safety level is proposed to help engineers design a safe and economical pile wall.展开更多
This paper studies a substitution secant/finite difference （SSFD） method for solving large scale sparse unconstrained optimization problems. This method is a combination of a secant method and a finite difference me...This paper studies a substitution secant/finite difference （SSFD） method for solving large scale sparse unconstrained optimization problems. This method is a combination of a secant method and a finite difference method, which depends on a consistent partition of the columns of the lower triangular part of the Hessian matrix. A q-superlinear convergence result and an r-convergence rate estimate show that this method has good local convergence properties. The numerical results show that this method may be competitive with some currently used algorithms.展开更多
This paper proposes a filter secant method with nonmonotone line search for non-linearequality constrained optimization.The Hessian of the Lagrangian is approximated using the BFGSsecant update.This new method has mor...This paper proposes a filter secant method with nonmonotone line search for non-linearequality constrained optimization.The Hessian of the Lagrangian is approximated using the BFGSsecant update.This new method has more flexibility for the acceptance of the trial step and requires lesscomputational costs compared with the monotone one.The global and local convergence of the proposedmethod are given under some reasonable conditions.Further,two-step Q-superlinear convergence rateis established by introducing second order correction step.The numerical experiments are reported toshow the effectiveness of the proposed algorithm.展开更多
This paper studies a family of the local convergence of the improved secant methods for solving the nonlinear equality constrained optimization subject to bounds on variables. The Hessian of the Lagrangian is approxim...This paper studies a family of the local convergence of the improved secant methods for solving the nonlinear equality constrained optimization subject to bounds on variables. The Hessian of the Lagrangian is approximated using the DFP or the BFGS secant updates. The improved secant methods are used to generate a search direction. Combining with a suitable step size, each iterate switches to trial step of strict interior feasibility. When the Hessian is only positive definite in an affine null subspace, one shows that the algorithms generate the sequences converging q-linearly and two-step q-superlinearly. Yhrthermore, under some suitable assumptions, some sequences generated by the algorithms converge locally one-step q-superlinearly. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithms.展开更多
The new type of torsional angle transformer-quarter-wave length hyperbolic secant taper was derived in which the maximum point of surface shear stress is inexistent. The half-wave length complex horns can be composed ...The new type of torsional angle transformer-quarter-wave length hyperbolic secant taper was derived in which the maximum point of surface shear stress is inexistent. The half-wave length complex horns can be composed on the basis of the new taper and quarter-wave taper which may be cylindrical exponential, sirnilar-catenary or similar-cosinc. The resonant frequency and angle amplification ratio of some complex transformers were measured. The experimental results basically agree with the theoretical values. The complex horns have a serics of good characteristics, such as the amplification ratio is very large, the changc of input impedance with the deviation of the frequency is little, the maximum output ultrasonic power is much higher than that of other types of the same material. Therefore, they are very useful horns in high-intensity ultrasonic field.展开更多
基金Supported by the National Natural Science Foundation of China (10871178)the Natural Science Foundation of Zhejiang Province of China (Y606154)Foundation of the Education Department of Zhejiang Province of China (20071362)
文摘A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.
基金the financial support from the Natural Science Basic Research Program of Shaanxi(No.2019JQ857)the Doctoral Scientific Research Foundation of Xi’an Polytechnic University(No.BS201863)+1 种基金the China Textile Industry Federation Science and Technology Guidance Project(No.2019002)the Tianyuan Open Fund Project(No.tywl2019-14)。
文摘To predict the Elasto-Plastic Behaviors(EPBs)of aligned inclusions reinforced composites,this paper develops an interpolative Mori-Tanaka/Double-Inclusion(MT-DI)homogenization model with the secant formulation,and gives the numerical implementation algorithms of the developed MT-DI model with the secant formulation.The Finite Element(FE)homogenization method is implemented to provide the "exact" EPBs of the composites and thus validate the MT-DI model with the secant formulation.The MT-DI model with the 2 nd-order secant formulation is validated to provide the more accurate predictions,while the MT-DI model with the 1 st-order secant formulation always gives the stiffer predictions.The results show that using the macro-stress and macrostrain as the inputs,the MT-DI model with the secant formulation gives the identical predictions.The predictions of the MT-DI model with the secant formulation vary between those of the MT and DI models with the secant formulation.Meanwhile,the MT-DI model with the secant formulation does not predict the accurate EPBs for the phases of the composites.
文摘Deflection is a significant indicator of bridge’s strength and its whole stiffness, so the research on deflection measurement is an important aspect of bridge health monitoring. There have existed many measurement methods of bridge deflection so far, while inclination method is gradually catching more and more attention for its fair obviously comprehensive advantages. However, the inclination method at present focuses on measuring the rotation of bridge’s section at testing point, that is, the tangent angle of deflection curve. With the tangent angle, the deflection curve can be determined by the methods of curve fitting or (and) integration or conjugate beam. The methods mentioned above, are not only complicated in calculation, but also bad in accuracy. The deflection measurement method proposed by this paper is based on measuring the inclination of two points initiatively in horizontal line, that is, the secant angle of the deflection curve, and on the simple triangle function operation. The proposed method is simple in theory, but good in accuracy for either static or dynamic load. The numerical simulation suggests that the error of the proposed method is less than 1%.
基金Supported by the Qianjiang Rencai Project Foundation of Zhejiang Province (J20070288)
文摘In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.
基金Supported by Science and Technology Foundation of Shanghai Higher Education
文摘The secant methods discussed by Fontecilla (in 1988) are considerably revised through employing a trust region multiplier strategy and introducing a nondifferentiable merit function. In this paper the secant methods are also improved by adding a dogleg typed movement which allows to overcome a phenomena similar to the Maratos effect. Furthermore, these algorithms are analyzed and global convergence theorems as well as local superlinear convergence rate are proved.
基金supported by National Natural Science Foundation of China(11771393,11371320,11632015)Zhejiang Natural Science Foundation(LZ14A010002,LQ18A010008)Scientific Research Fund of Zhejiang Provincial Education Department(FX2016073)
文摘Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we also provide an error estimate that matches the convergence order of the two-step secant method.At last,we give an application of the proposed theorem.
文摘A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x^＊, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.
文摘In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.
文摘In this paper, we present m time secant like multi projection algorithm for sparse unconstrained minimization problem. We prove this method are all q superlinearly convergent to the solution about m≥1 . At last, we from some numerical results, discuss how to choose the number m to determine the approximating matrix properly in practical use.
文摘In this paper, we discuss the relationship between the sparse symmetric Broyden (SPSB) method [1, 2] and m-time secant-like multi-projection (SMP) method [3] and prove that when m goes to infinity, the SMP method is corresponding to the SPSB method.
基金the Research Council of Semnan University for its support
文摘By making a convex combination of the modified secant equations proposed by Yuan and Wei et al.,a hybrid secant equation and also,a modified BFGS algorithm is proposed.The hybridization parameter is effectively computed using the available information of recent iterations.Under proper conditions,it is shown that the proposed algorithm is globally,locally and superlinearly convergent.By using the performance profile introduced by Dolan and Mor6,a comparison between the implementations of the proposed algorithm and two efficient modified BFGS algorithms proposed by Yuan and Wei et al.,on a set of unconstrained optimization test problems from the CUTEr collection,is done.Numerical results demonstrating the efficiency of the proposed modified BFGS algorithm are reported.
基金supported by the National Science Foundation under Grant No.10871130, the Ph.D. Foundation of Chinese Education Ministry under Grant No.20093127110005the Shanghai Leading Academic Discipline Project under Grant No.T0401
文摘This paper formulates and analyzes a line search method for general nonlinear equalityconstrained optimization based on filter methods for step acceptance and secant methods for searchdirection.The feature of the new algorithm is that the secant algorithm is used to produce a searchdirection,a backtracking line search procedure is used to generate step size,some filtered rules areused to determine step acceptance,second order correction technique is used to reduce infeasibility andovercome the Maratos effect.Global convergence properties of this method are analyzed:under mildassumptions it is showed that every limit point of the sequence of iterates generated by the algorithmis feasible,and that there exists at least one limit point that is a stationary point for the problem.Moreover,it is also established that the Maratos effect can be overcome in our new approach by addingsecond order correction steps so that fast local superlinear convergence to a second order sufficient localsolution is achieved.Finally,the results of numerical experiments are reported to show the effectivenessof the line search filter secant method.
基金the National Natural Science Foundation of China(Grant Nos.51978040 and 51378054)the National Basic Research Program of China(973 program,No.2015CB057800)for supporting this research.
文摘The failure to achieve minimum design overlap between secant piles compromises the ability of a structure to perform as designed,resulting in water leakage or even ground collapse.To establish a more realistic simulation and provide guidelines for designing a safe and cost-effective secant-pile wall,a three-dimensional model of a secant pile,considering the geometric imperfections of the diameter and direction of the borehole,is introduced.An ultrasonic cross-hole test was performed during the construction of secant piles in a launching shaft in Beijing,China.Based on the test results,the statistical characteristics of the pile diameters and orientation parameters were obtained.By taking the pile diameter D,inclination angleβ,and azimuth angleαas random variables,Monte Carlo simulations were performed to discuss the influence of different design parameters on the probability density functions of the overlap of secant piles.The obtained results show that the randomness of the inclination angle and pile diameter can be well described by a normal distribution,whereas the azimuth angle is more consistent with a uniform distribution.The integrity of the secant-pile wall can be overestimated without considering the uncertainty of geometric imperfections.The failure of the secant-pile wall increases substantially with increasing spatial variability in drilling inclination and diameter.A design flowchart for pile spacing under the target safety level is proposed to help engineers design a safe and economical pile wall.
基金Supported by the National Natural Science Foundation of China （No.10471015） and the State Foundation of Ph.D Units of China （No.20020141013）
文摘This paper studies a substitution secant/finite difference （SSFD） method for solving large scale sparse unconstrained optimization problems. This method is a combination of a secant method and a finite difference method, which depends on a consistent partition of the columns of the lower triangular part of the Hessian matrix. A q-superlinear convergence result and an r-convergence rate estimate show that this method has good local convergence properties. The numerical results show that this method may be competitive with some currently used algorithms.
基金supported by the National Science Foundation of China under Grant No. 10871130the Ph.D. Foundation under Grant No. 20093127110005+1 种基金the Shanghai Leading Academic Discipline Project under Grant No. S30405the Shanghai Finance Budget Project under Grant Nos. 1139IA0013 and 1130IA15
文摘This paper proposes a filter secant method with nonmonotone line search for non-linearequality constrained optimization.The Hessian of the Lagrangian is approximated using the BFGSsecant update.This new method has more flexibility for the acceptance of the trial step and requires lesscomputational costs compared with the monotone one.The global and local convergence of the proposedmethod are given under some reasonable conditions.Further,two-step Q-superlinear convergence rateis established by introducing second order correction step.The numerical experiments are reported toshow the effectiveness of the proposed algorithm.
基金supported by the partial supports of the National Science Foundation under Grant No.10871130the Ph.D. Foundation under Grant No.20093127110005 of Chinese Education Ministry
文摘This paper studies a family of the local convergence of the improved secant methods for solving the nonlinear equality constrained optimization subject to bounds on variables. The Hessian of the Lagrangian is approximated using the DFP or the BFGS secant updates. The improved secant methods are used to generate a search direction. Combining with a suitable step size, each iterate switches to trial step of strict interior feasibility. When the Hessian is only positive definite in an affine null subspace, one shows that the algorithms generate the sequences converging q-linearly and two-step q-superlinearly. Yhrthermore, under some suitable assumptions, some sequences generated by the algorithms converge locally one-step q-superlinearly. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithms.
文摘The new type of torsional angle transformer-quarter-wave length hyperbolic secant taper was derived in which the maximum point of surface shear stress is inexistent. The half-wave length complex horns can be composed on the basis of the new taper and quarter-wave taper which may be cylindrical exponential, sirnilar-catenary or similar-cosinc. The resonant frequency and angle amplification ratio of some complex transformers were measured. The experimental results basically agree with the theoretical values. The complex horns have a serics of good characteristics, such as the amplification ratio is very large, the changc of input impedance with the deviation of the frequency is little, the maximum output ultrasonic power is much higher than that of other types of the same material. Therefore, they are very useful horns in high-intensity ultrasonic field.
