标准形式 [ 编辑 ]

العربية বাংলা فارسی français עברית हिन्दी 新加坡简体 臺灣正體 获取缩短的url 引用本页 打印/导出 维基共享资源 维基百科，自由的百科全书 单纯形法（ simple x algorithm）在数学优化领域中常用于 线性规划 问题的 数值求解 ，由 喬治·伯納德·丹齊格 发明。 下山单纯形法 （nelder-mead method）与单纯形法名称相似，但二者关联不大。该方法由nelder和mead于1965年发明，是用于优化多维无约束问题的一种数值方法，属于更普遍的 搜索算法 的类别。这两种方法都使用了 单纯形 的概念。单纯形是 {displaystyle n+1} 个 顶点 的 凸包 ，是一个 多胞体 ：直线上的一个线段，平面上的一个三角形，三维空间中的一个 四面体 等等，都是单纯形。 标准形式[ 编辑 ] 假设有n个 变量 和m个 约束 。线性规划的标准形式如下： {displaystyle {egin{aligned}&&max sum limits _{1leq kleq n}{{{c}_{k}}{{x}_{k}}}\&s. T. &sum limits _{1leq kleq n}{{{a}_{1,k}}{{x}_{k}}}leq {{b}_{1}},\&&sum limits _{1leq kleq n}{{{a}_{2,k}}{{x}_{k}}leq {{b}_{2}},}\&. \&&sum limits _{1leq kleq n}{{{a}_{m,k}}{{x}_{k}}}leq {{b}_{m}}\&&{{x}_{1}},{{x}_{2,}}. ,{{x}_{n}}geq 0end{aligned}}} 所有其他形式的线性规划方程组都可以按照下列方式转化成标准形式： 目标函数 并非最大化：将所有 约束条件中存在大于或等于约束：将约束两边取负。 约束条件中存在 等式 ：将其拆分为两个 不等式 （一个大于等于，一个小于等于） 有的变量没有非负约束：加入新变量 松弛形式[ 编辑 ] 可以将标准形式的线性规划转化为松弛形式，以方便运算。 在原来n个变量，m个约束的线性规划中，加入m个新的变量，将原来的不等式化为等式： {displaystyle {{x}_{n+j}}={{b}_{j}}-sum limits _{1leq kleq n}{{{a}_{j,k}}{{x}_{k}}}} 当然，此时 依然成立。 {displaystyle {{x}_{1}},{{x}_{2}},. ,{{x}_{n}}} 这些变量称为非基变量，它们构成的 集合 记为n。将 {displaystyle {{x}_{n+1}},{{x}_{n+2}},. ,{{x}_{n+m}}} 这些变量称为基变量，它们构成的集合记为b。简单地理解，非基变量能够由基变量唯一确定。 在这样的定义下，线性规划的松弛形式可以写为如下形式： {displaystyle {egin{aligned}&max sum limits _{kin n}{{{c}_{k}}{{x}_{k}}}\&s. T. \&orall 1leq ileq n+m,{{x}_{i}}geq 0\&orall jin b,{{x}_{j}}={{b}_{j}}-sum limits _{kin n}{{{a}_{j,k}}{{x}_{k}}}\end{aligned}}} 因此，线性规划的松弛形式可以由 唯一确定， {displaystyle n,b} 是 整数 集合，分别表示非基变量集合以及基变量集合。 转轴操作[ 编辑 ] 转轴操作是单纯形法中的核心操作，其作用是将一个基变量与一个非基变量进行互换。可以将转轴操作理解为从 单纯形 上的一个 顶点 走向另一个顶点。 {displaystyle {{x}_{e}}} 属于n（非基变量），执行转轴操作pivot(d,e)之后， 将变为基变量。 具体地说，一开始我们有 移项 ，得 {displaystyle a_{d,e}x_{e}=b_{d}-sum limits _{kin n,k eq e}a_{d,k}x_{k}-{x}_{n+d}} ，我们有 {displaystyle {{x}_{e}}={rac {{b}_{d}}{{a}_{d,e}}}-(sum limits _{kin n,k eq e}{{rac {{a}_{d,k}}{{a}_{d,e}}}{{x}_{k}})}-{rac {1}{{a}_{d,e}}}{{x}_{n+d}}}.

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Write the initial tableau of simplex method. The initial tableau of simplex method consists of all the coefficients of the decision variables of the original problem and the slack, surplus and artificial variables added in second step (in columns, with p0 as the constant term and pi as the coefficients of the rest of xi variables), and constraints (in rows). The cb column contains the coefficients of the variables that are in the base. The first row consists of the objective function coefficients, while the last row contains the objective function value and reduced costs zj - cj. The last row is calculated as follows: zj = σ(cbi·pj) for i = 1.

Blog imprint the simplex algorithm is a popular method for numerical solution of the linear programming problem. The algorithm solves a problem accurately within finitely many steps, ascertains its insolubility or a lack of bounds. It was created by the american mathematician george dantzig in 1947. Since that time it has been improved numerously and become one of the most important methods for linear optimization in practice.

Download wolfram notebook a simplex, sometimes called a hypertetrahedron (buekenhout and parker 1998), is the generalization of a tetrahedral region of space to dimensions. The boundary of a -simplex has 1-faces ( polytope edges ), and -faces, where is a binomial coefficient. -dimensional simplex can be denoted using the schläfli symbol. The simplex is so-named because it represents the simplest possible polytope in any given space. The content (i. E. , hypervolume) of a simplex can be computed using the cayley-menger determinant. In one dimension, the simplex is the line segment. In two dimensions, the simplex is the convex hull of the equilateral triangle. In three dimensions, the simplex is the convex hull of the tetrahedron.

The simplex package is a collection of routines for linear optimization using the simplex algorithm as a whole, and using only certain parts of the simplex algorithm. In addition to the routines feasible, maximize, and minimize, the simplex package provides routines to assist the user in carrying out the steps of the algorithm one at a time: setting up problems, finding a pivot element, and executing a single pivot operation. To directly obtain a numerical solution to a linear program, it is recommended that you use the optimization[lpsolve] command, which is more efficient for this purpose. Each command in the simplex package can be accessed by using either the long form or the short form of the command name in the command calling sequence.