避免"分类讨论"的几种策略

分类讨论既是一种重要的数学思想方法,又是一种重要的解题策略,在数学解题中有着广泛的应用.但分类讨论时,一般过程都较为冗长、繁琐,且极易在完备性上造成失误.因此,在分类之前应有意识地调整思维策略,尽量地避免分类讨论,以简化或优化解题过程,达到简捷解题的目的.本文介绍几种避免分类讨论的解题策略.……     

简化和避免分类讨论的方法

<正>在重视分类讨论思想应用的基础上,应防止见参数就讨论的轻率做法,能整体解决的就不必分类讨论,树立辩证的解题观点,使分类讨论用的更为合理.简化和避免分类讨论的优化策略主要有以下几种:(1)直接回避如应用反证法、求补法、消参数等方法,有时可避开繁琐讨论.(2)变更主元如分离参数变参置换、构造以讨论对象为变量的函数等形式,解题时可     

分类讨论思想在解数列题中的应用

<正>分类讨论思想是一种常见的数学解题思想,常应用于数列,函数与导数有关的考题中.既然分类,就应有分类标准与依据,若能合理进行,则会起到事半功倍之效,不然,则会陷入讨论困境.本文就数列解题中涉及的分类讨论思想进行归纳整理,得到常见的三种分类题型.类型一、按项数奇偶进行分类例1(人教B版教材必修5习题)在数     

处理导数问题的几种方法

<正>导数问题灵活多变,难度性大,常用的解题方法有分类讨论,分离参数.但是,在大多情况下,分离参数后函数或导函数形式复杂而无法求出函数的最值,解题过程往往是有始无终;若不分离,直接分类讨论,分类层次较多又会出现重复或者漏解.面对这样的两难问题.笔者给出解决此类题的四个妙招.一、特值代入,缩小讨论范围     

挖掘题目内涵 寻找解题突破口

有些题目不是很容易看出解题思路的,而是要结合题目条件和结论,充分利用已有的知识点和解题方法,深挖题目内涵,实行转化化归,并把数形结合思想、函数和方程思想、分类讨论思想等进行有机结合,巧妙变换,寻找解题突破口.一、利用抽象函数关系,巧妙变换解题     

分类思想在求一次函数解析式中的应用

<正>在解答某些数学题时,有时会遇到多种情况,需要对各种情况加以分类,并逐类求解,然后综合得解,这就是分类讨论思想.它是数学中一种重要的解题方法,不仅能帮助我们顺利地解决一些问题,也能培养我们的观察能力和全面思考问题的能力.例1一次函数y=kx+b的自变量x的取值范围是-3≤x≤6,相应函数值的取值范围是-5≤y≤-2,求这个函数的解析式.     

讨论与否 存乎一心

众所周知,分类讨论是一种重要的数学思想和解题策略,在中学数学学习中占有非常重要的位置,当然,利用分类讨论解题时,需要对问题进行分门别类的讨论,有时就难免使问题的解决过程变得繁琐冗长．因此,我们又希望避免解题过程中的分类讨论．     

我们是如何陷入分类讨论的?

分类讨论是一种重要的数学思想和解题策略,在中学数学学习中有重要的位置.当然,由于分类讨论,也难免使得问题的解决过程变得繁杂冗长.因此,我们又希望避免解题过程中的分类讨论.事实上,解决某些数学问题,之所以要分类讨论,常常是囿于我们所选择的解题视角,而不是问题本身的缘故.     

数形结合能优化分类讨论过程

为什么要分类讨论和怎样分类讨论,即分 类标准的选取是解题中很难把握的环节;下面 介绍一种既能较易找准分类标准的切入点,又 能优化解题过程,降低解题难度的方法--数 形结合. 一、优化含参不等式中的分类讨论 例1 解关于x的不等式 解 由题意可     

不走寻常路 不一样的精彩——例谈避免分类讨论的解题策略

分类讨论是一种重要的数学思想方法,对于其中有些问题,因为分类讨论论述较长,讨论过程往往十分烦琐,而且容易讨论不完整造成解题失误.但如果我们把学习数学注入"生命"的灵动,注意克服思维定势,力求简化分类讨论甚至避免分类讨论,以求解法的简捷,从而提高解题速度和解题的准确性.因此,我们提倡在熟悉和掌握分类讨论思想的同时,要注意如何避免讨论,本文从几个方面论述,避免讨论的对策,以供参考.一、换个视角更换主元避免分类讨论     

基于不等式约束的一类新的增广lagrangian函数

本文提出了一类新的增广lagrangian函数,并证明了它的稳定点、整体极小点与原约束问题KKT点、整体极小点有1-1对应关系,增广lagrangian函数的局部极小点为原问题的局部极小点.     

Inverse Eigenvalue Problem in Structural Dynamics Design

1 Introduction Structural dynamics design is to design a structure subject to the dynamic characteristics re- quirement, i.e., determine physical and geometrical parameters such that the structure has the given frequencies and (or) mode shapes. This problem often arises in engineering connected with vibration. Recently, Joseph [1], Li et al. [2,3] converted the structural dynamics design to the following inverse eigenvalue problem. GIEP Let x = (x1, , xm)T , and let A(x) and B(x) be real n…     

Solución de la cinemática directa de la plataforma Gough-Stewart general usando un algoritmo híbrido de optimización

The direct kinematics problem for parallel robots can be stated as follows: given values of the joint variables, the corresponding Cartesian variable values, the pose of the end-effector, must be found. Most of the times the direct kinematics problem involves the solution of a system of non-linear equations. The most efficient methods to solve such kind of equations assume convexity in a cost function which minimum is the solution of the non-linear system. In consequence, the capacity of such methods depends on the knowledge about an starting point which neighboring region is convex, hence the method can find the global minimum. This article propose a method based on probabilistic learning about an adequate starting point for the Dogleg method which assumes local convexity of the function. The proposed method efficiently avoids the local minima, without need of human intervention or apriori knowledge, thus it shows a more robust performance than the simple Dogleg method or other gradient based methods. To demonstrate the performance of the proposed hybrid method, numerical experiments and the respective discussion are presented. The proposal can be extended to other structures of closed-kinematics chains, to the general solution of systems of non-linear equations, and to the minimization of non-linear functions.     

关于DC函数最小化邻近点算法的注记

本文证明DC函数最小化问题邻近点算法的一个收敛性定理,并对此问题提出一类非精确邻近点算法．     

Clifford 分析中超正则函数向量的一个边值问题

By the Plemelj formula and the compressed fixed point theorem,this paper discusses a kind of boundary value problem for hypermonogenic function vectors in Clifford analysis.And the paper proves the existence and uniqueness of the solution to the boundary value problem for hypermonogenic function vectors in Clifford analysis.     

分布参数最优控制的边界元共轭梯度算法

研究了一类线性椭圆型分布参数最优控制问题的数值解算法.得到最优控制对应的最优性方程组,在凸性条件下,证明了最优控制的唯一存在性问题.将最优控制问题化为以控制函数和状态函数为局中人的递阶式(Stackelberg)非合作对策问题,其平衡点是最优控制的解.进一步得到求平衡点的边界元共轭梯度算法.最后,研究算法中边界元离散的误差估计,以算例验证该算法.     

Stability Analysis of Gradient-Based Neural Networks for Optimization Problems

The paper introduces a new approach to analyze the stability of neural network models without using any Lyapunov function. With the new approach, we investigate the stability properties of the general gradient-based neural network model for optimization problems. Our discussion includes both isolated equilibrium points and connected equilibrium sets which could be unbounded. For a general optimization problem, if the objective function is bounded below and its gradient is Lipschitz continuous, we prove that (a) any trajectory of the gradient-based neural network converges to an equilibrium point, and (b) the Lyapunov stability is equivalent to the asymptotical stability in the gradient-based neural networks. For a convex optimization problem, under the same assumptions, we show that any trajectory of gradient-based neural networks will converge to an asymptotically stable equilibrium point of the neural networks. For a general nonlinear objective function, we propose a refined gradient-based neural network, whose trajectory with any arbitrary initial point will converge to an equilibrium point, which satisfies the second order necessary optimality conditions for optimization problems. Promising simulation results of a refined gradient-based neural network on some problems are also reported.     

Reduction of a Minimization Problem of a Separable Convex Function Under Linear Constraints to a Fixed Point Problem

The paper is devoted to studying a constrained nonlinear optimization problem of a special kind. The objective functional of the problem is a separable convex function whose minimum is sought for on a set of linear constraints in the form of equalities. It is proved that, for this type of optimization problems, the explicit form can be obtained of a projection operator based on a generalized projection matrix. The projection operator allows us to represent the initial problem as a fixed point problem. The explicit form of the fixed point problem makes it possible to run a process of simple iteration. We prove the linear convergence of the obtained iterative method and, under rather natural additional conditions, its quadratic convergence. It is shown that an important application of the developed method is the flow assignment in a network of an arbitrary topology with one pair of source and sink.     

Interior proximal methods for quasiconvex optimization

A generalized proximal point algorithm for the minimization of a nonconvex function on a feasible set is investigated. It is known that if the objective function of the given problem is (lower semicontinuous, proper and) convex, well-definedness of the method as well as convergence of the generated iterates, being the solutions of better conditioned and uniquely solvable subproblems, are known. The present paper contributes to the discussion of the methods’ behaviour when the objective is not convex. This gives rise to questions, among others, of well-definedness and convergence of the generated sequence.     

Optimization with Variable Sets of Constraints and an Application to Truss Design

We discuss the minimization of a continuous function on a subset of Rⁿ subject to a finite set of continuous constraints. At each point, a given set-valued map determines the subset of constraints considered at this point. Such problems arise e.g. in the design of engineering structures.After a brief discussion on the existence of solutions, the numerical treatment of the problem is considered. It is briefly motivated why standard approaches generally fail. A method is proposed approximating the original problem by a standard one depending on a parameter. It is proved that by choosing this parameter large enough, each solution to the approximating problem is a solution to the original one. In many applications, an upper bound for this parameter can be computed, thus yielding the equivalence of the original problem to a standard optimization problem.The proposed method is applied to the problem of optimally designing a loaded truss subject to local buckling conditions. To our knowledge this problem has not been solved before. A numerical example of reasonable size shows the proposed methodology to work well.