The multiplex method for linear programming
The multiplex method for linear programming
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线性规划的复用方法
DOI：
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发表时间：
1958
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通讯作者：
R. Frisch
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文献类型：
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作者：
R. Frisch
1. 在标准形式下，线性规划问题可以表述如下。满足m个线性无关的线性方程。因此，自由度的数量是n至少有一种方法可以用n个基变量的集合来表示所有的变量，它们之间是线性无关的。设为这样一个基集。然后方程可以写成标准形式xj = &jo + s hkxk (j，其中bj0和bjk是常数。显然,我呢?k =j bjQ = 0, bjk = 0，否则t：若对所有j = 1,2，…n+m，我们得到一个线性相关的方程组，但是如果我们只取（1.3）对于1j我们得到一个线性无关的方程组。更准确地说：如果系数bj0和bjk有任何值，则m个形式为（1.3）的方程，我们考虑线性偏好函数f = P0+PwV*+Pvtv+ ？+ Puflu > ?（L5）其中pk (k = 0, u, v…W)是任意给定的常数为正、负或零。如果我们假设（1.5）中只出现基变量，则不会限制通用性。实际上，当所有变量都用（1.3）表示为基变量时，所有n+m个变量的任何线性函数都将采用（1.5）的形式。线性规划问题是确定在两组条件下使（1.5）最大化的一组或几组变量值的问题。首先是等式（1.3），其次是由不等式表达的非负性条件（我用倒括号）……表示“不包括”。
1. The problem In standard form the linear programming problem can be formulated as follows. satisfying m linearly independent linear equations. The number of degrees of freedom is consequently n and it is always possible at least in one way to express all the variables in terms of a set of n basis variables, linearly indepen dent amongst themselves. Let be such a basis set. The equations may then be written in the standard form xj = &jo + s hkxk (j where the bj0 and bjk are constants. Obviously i?k =j bjQ = 0 and bjk = otherwise t: If the equations (1.3) are taken for all j = 1,2,...,n+m, we get a system of equations that are linearly dependent, but if we take (1.3) only for1 j we get a system of equations that are linearly independent. More precisely : If the coefficients bj0 and bjk have any values whatsoever, m equations of the form (1.3) for We consider a linear preference function f = P0+PwV*+Pvtv+ ? +Puflu> ? (L5) where the pk (k = 0, u, v ... w) are any given constants positive, negative or zero. It does not restrict generality if we assume that only basis variables occur in (1.5). Indeed, any linear function of all the n+m variables will assume the form (1.5) when all the variables are expressed in terms of the basis variables by means of (1.3). The linear programming problem is the problem of determining that one or those sets of values of the variables that will maximize (1.5) subject to two sets of conditions. In the first place the equations (1.3), and in the second place the non-negativity conditions expressed by the inequalities iVe use the inverted parenthesis)...(to denote "exclusion of".