一类非单调曲线搜索方法及其收敛性

本文提出一类求解无约束优化问题的非单调曲线搜索方法, 在较弱条件下证明了其收敛性.该算法有如下特点:(1)采用曲线搜索方法, 在每步迭代时同时确定下降方向和步长;(2)采用非单调搜索技巧, 产生较大的迭代步长, 降低了算法的计算量;(3)利用当前和前面迭代点的信息产生下降方向, 无需计算和存储矩阵, 适于求解大型优化问题.     

“在”与“过”的差别

<正>《数学课程标准》及《高考考试说明》中要求学生能利用导数研究函数的单调性,会求函数的单调区间,会用导数函数的最值和极值.作为基础知识的导数的几何意义中,求曲线"在"某点处的切线和"过"某点的切线一类问题,让学生陷入了迷糊状态.下面举例来说明.例1曲线y=x³+x+1在点P(1,3)处的切线方程为___.解因为P(1,3)在曲线上,在该点处的切     

一道高考题的推广、类比拓展与证明

以下是2014年北京卷文科的一道高考题:已知函数f(x)=2x3-3x.(1)求f(x)在区间[-2,1]上的最大值;(2)若过点P(1,t)存在3条直线与曲线y=f(x)相切,求t的取值范围;(3)问过点A(-1,2),B(2,10),C(0,2)分别存在几条直线与曲线y=f(x)相切?(只需写出结论)本题考查了函数的导数题型.对于导数问题,高考重点考查两方面的内容:(1)函数的单调性;     

一个解不等式约束优化问题的初始点任意而不需罚函数的SQP算法(英文)

本文构造了一解不等式约束优化问题的非单调SQP方法 ,与类似的算法比较 ,它有以下特点 :( 1 )初始点任意 ,并不用罚函数 ;( 2 )有限步后必产生可行点 ;( 3)在每次迭代 ,只需解一个二次规划子问题 ;( 4)不需要严格互补条件 ,在较弱的条件下 ,算法超线性收敛 .     

含单参数的导函数符号问题

<正>利用导数证明不等式或求参数范围问题是近几年高考的一种热点题型,而解这类问题的真正难点是判断或讨论含单参数导函数的符号问题,本文结合具体实例阐述解这类问题的四种途径,仅供参考.一、依已知中参数的范围,去参后判断符号例1已知函数f(x)=(a+1)lnx+ax2+1,(1)讨论函数f(x)的单调性;(2)设a≤     

非单调多步曲线搜索方法的收敛性

提出一种求解无约束优化问题的非单调多步曲线搜索方法.此方法具有如下特点:(1)算法在产生下一个迭代点时不仅利用了当前迭代点的信息,而且还可能利用前m个迭代点的信息.这就是多步法;(2)下降方向和步长同时确定,而不是先找到方向,再由线性搜索寻找步长.这就是曲线搜索技术;(3)采用非单调搜索技巧.在较弱的条件下,我们证明了此方法的收敛性.     

曲线中的切线问题

导数是解决函数的单调性、最值、不等式证明等问题的有力工具,其应用相当广泛,因而是每年高考考查的重点与热点,但考生在这里失分较多,利用导数求曲线的切线方程是导数的重要应用之一,本文对此问题进行探索研究,归纳总结出了几种常见问题,供广大教师和同学们参考.1.给定切点的曲线切线问题例1求曲线y=xex+2x+1在点(0,1)处的切线方程.解因为点(0,1)在曲线y=xex+2x+1     

约束优化问题的一类光滑罚算法的全局收敛特性

对约束优化问题给出了一类光滑罚算法.它是基于一类光滑逼近精确罚函数 l_p(p\in(0,1]) 的光滑函数 L_p 而提出的.在非常弱的条件下, 建立了算法的一个摄动定理, 导出了算法的全局收敛性.特别地, 在广义Mangasarian-Fromovitz约束规范假设下, 证明了当 p=1 时, 算法经过有限步迭代后, 所有迭代点都是原问题的可行解; p\in(0,1) 时,算法经过有限迭代后, 所有迭代点都是原问题可行解集的内点.     

约束优化问题的一类光滑罚算法的全局收敛特性（英文）

对约束优化问题给出了一类光滑罚算法.它是基于一类光滑逼近精确罚函数l_p(p∈(0,1])的光滑函数L_p而提出的.在非常弱的条件下,建立了算法的一个摄动定理,导出了算法的全局收敛性.特别地,在广义Mangasarian-Fromovitz约束规范假设下,证明了当p=1时,算法经过有限步迭代后,所有迭代点都是原问题的可行解;当p∈(0,1)时,算法经过有限迭代后,所有迭代点都是原问题可行解集的内点.     

2012年课标全国卷理科第21题另解

2012年新课标全国卷理科数学第21题为:已知函数f(x)满足f(x)=f′(1)ex-1-f(0)x+12x2;(1)求f(x)的解析式及单调区间;(2)若f(x)≥12x2+ax+b,求(a+1)b的最大值.本题是函数、导数和不等式的综合题,立意新颖.第(2)小问以参数处理为主要特征,以导数应     

优化方法解决无上层约束的线性双层规划问题

In this paper we propose an optimal method for solving the linear bilevel programming problem with no upper-level constraint. The main idea of this method is that the initial point which is in the feasible region goes forward along the optimal direction firstly. When the iterative point reaches the boundary of the feasible region, it can continue to go forward along the suboptimal direction. The iteration is terminated until the iterative point cannot go forward along the suboptimal direction and effective direction, and the new iterative point is the solution of the lower-level programming. An algorithm which bases on the main idea above is presented and the solution obtained via this algorithm is proved to be optimal solution to the bilevel programming problem. This optimal method is effective for solving the linear bilevel programming problem.     

Subspace Search Method for Quadratic Programming With Box Constraints

1.IntroductionInthispaper,weconsidertheproblemofminimizingaquadraticconvexprogrammingwithboxconstrainedvariables:Minf(x)s.t.x6fi(1'1)1,wherefi={xER":15xSu},f(x)=AX"Hx bTx,andHisannbynsymmetric.2positivedefinitematrix,andb,l,uaregivenconstantvectorsin...     

Superlinear Convergence of a Smooth Approximation Method for Mathematical Programs with Nonlinear Complementarity Constraints

<正>Mathematical programs with complementarity constraints(MPCC) is an important subclass of MPEC.It is a natural way to solve MPCC by constructing a suitable approximation of the primal problem.In this paper,we propose a new smoothing method for MPCC by using the aggregation technique.A new SQP algorithm for solving the MPCC problem is presented.At each iteration,the master direction is computed by solving a quadratic program,and the revised direction for avoiding the Maratos effect is generated by an explicit formula.As the non-degeneracy condition holds and the smoothing parameter tends to zero,the proposed SQP algorithm converges globally to an S-stationary point of the MPEC problem,its convergence rate is superlinear.Some preliminary numerical results are reported.     

Planar quasi-Newton algorithms for unconstrained saddlepoint problems

A new class of quasi-Newton methods is introduced that can locate a unique stationary point of ann-dimensional quadratic function in at mostn steps. When applied to positive-definite or negative-definite quadratic functions, the new class is identical to Huang's symmetric family of quasi-Newton methods (Ref. 1). Unlike the latter, however, the new family can handle indefinite quadratic forms and therefore is capable of solving saddlepoint problems that arise, for instance, in constrained optimization. The novel feature of the new class is a planar iteration that is activated whenever the algorithm encounters a near-singular direction of search, along which the objective function approaches zero curvature. In such iterations, the next point is selected as the stationary point of the objective function over a plane containing the problematic search direction, and the inverse Hessian approximation is updated with respect to that plane via a new four-parameter family of rank-three updates. It is shown that the new class possesses properties which are similar to or which generalize the properties of Huang's family. Furthermore, the new method is equivalent to Fletcher's (Ref. 2) modified version of Luenberger's (Ref. 3) hyperbolic pairs method, with respect to the metric defined by the initial inverse Hessian approximation. Several issues related to implementing the proposed method in nonquadratic cases are discussed.An earlier version of this paper was presented at the 10th Mathematical Programing Symposium, Montreal, Canada, 1979.     

On Global Convergence and Approximate Iteration of the Linear Approximation Method for Solving Variational Inequalities

This paper is concerned with the linear approximation method (i.e. the iterative method in which a sequence of vectors is generated by solving certain linearized subproblems) for solving the variational inequality. The global convergent iterative process is proposed by applying the continuation method, and the related problems are discussed. A convergent result is obtained for the approximation iteration (i.e. the iterative method in which a sequence of vectors is generated by solving certain linearized subproblems approximately).     

Piecewise polynomial Taylor methods for initial value problems

A class of explicit Taylor-type methods for numerically solving first-order ordinary differential equations is presented. The basic idea is that of generating a piecewise polynomial approximating function, with a given order of differentiability, by repeated Taylor expansion. Sharp error bounds for the approximation and its derivatives are given along with a stability analysis.This work was supported by the United States Atomic Energy Commission.     

The combination of collocation,finite difference,and multigrid methods for solution of the two‐dimensional wave equation

In this article, we apply compact finite difference approximations of orders two and four for discretizing spatial derivatives of wave equation and collocation method for the time component. The resulting method is unconditionally stable and solves the wave equation with high accuracy. The solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. We employ the multigrid method for solving the resulted linear system. Multigrid method is an iterative method which has grid independently convergence and solves the linear system of equations in small amount of computer time. Numerical results show that the compact finite difference approximation of fourth order, collocation and multigrid methods produce a very efficient method for solving the wave equation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

关于半定规划的一种宽邻域不可行内点算法的注记

针对半定规划的宽邻域不可行内点算法, 将牛顿法和预估校正法进行结合, 构造出适当的迭代方向, 提出一个修正的半定规划宽邻域不可行内点算法, 并在适当的假设条件下, 证明了该算法具有O(\sqrt{n}L)的迭代复杂界．最后利用Matlab编程, 给出了基于KM方向和NT方向的数值实验结果．     

SUBSPACE SEARCH METHOD FOR A CLASS OF LEAST SQUARES PROBLEM

1. IntroductionNow the least squares problem is considered as follows:1Mid r(x,y) ~ SllAx By ~ bll' s.t. x 2 0 (1.1)where A E Rm",, B E R"q, and b E Re are given constant matrices and vectors,respectively.These problems arise in many areas of applications, such as scientific and engineering computing, physics, statistics, flited curve, economic, mathematical programming,social science, and as a component part of some large computation problem, as anexample, a nonlinear least squares pr…     

A modified Newton’s method for best rank-one approximation to tensors

In this paper, a modified Newton’s method for the best rank-one approximation problem to tensor is proposed. We combine the iterative matrix of Jacobi-Gauss-Newton (JGN) algorithm or Alternating Least Squares (ALS) algorithm with the iterative matrix of GRQ-Newton method, and present a modified version of GRQ-Newton algorithm. A line search along the projective direction is employed to obtain the global convergence. Preliminary numerical experiments and numerical comparison show that our algorithm is efficient.