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# 数学代写|优化理论代写Optimization Theory代考|Problems with explicitly bounded variables

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## 数学代写|优化理论代写Optimization Theory代考|Problems with explicitly bounded variables

There are several considerations that go into an implementation of the Simplex Algorithm on a computer. For example, the constraints in a given model will not necessarily all be equations, or all be inequalities. Further, many activity levels may have a lower bound different from 0 and an upper bound that is finite. It is a routine exercise to convert such a linear program into standard form. However, from a computational standpoint, it is inefficient to consider the explicit representation of upper-bounded variables as separate inequality constraints. We shall see that with suitable modification of the Simplex Algorithm, it is unnecessary to carry out such transformations.
To appreciate the distinction between explicit and implicit representations of upper-bounded variables consider the following two systems
\begin{aligned} & x_1+x_2+\cdots+x_n=1 \ & 0 \leq x_j \leq 1, \quad j=1, \ldots, n, \end{aligned}
and
\begin{aligned} & x_1+x_2+\cdots+x_n=1 \ & x_j \geq 0, \quad j=1, \ldots, n . \end{aligned}
In (7.1) the variables $x_j$ all have explicit upper-bound constraints. In (7.2) the variables are bounded above, but the upper-bound constraints $x_j \leq 1$ are implicit; they are a consequence of the other constraints. Needless to say, there are LP formulations in which some of the variables are explicitly bounded while others are implicitly bounded.

## 数学代写|优化理论代写Optimization Theory代考|Lower and upper bounds on variables

Consider an LP with $m$ equality constraints in $n$ variables having explicit lower and upper bounds that are not necessarily 0 and $\infty$ respectively.
$$\begin{array}{ll} \text { minimize } & c^{\mathrm{T}} x \ \text { subject to } & A x=b, \ & l_j \leq x_j \leq u_j \quad \text { for } j=1,2, \ldots, n . \end{array}$$
We will assume that each variable has at least one finite bound. Any variable, not satisfying this assumption must be a free variable and hence can be replaced by the difference of two nonnegative variables, thereby making the assumption hold. Furthermore, any linear program can be put in this form.

For the moment, let us assume that for $j=1,2, \ldots, n$ we have $l_j=0$, and every upper bound $u_j$ finite and positive. Such a problem can be put in standard form by adding slack variables $x_j^{\prime} \geq 0$ to the constraints $x_j \leq u_j$ as follows:
$$\begin{array}{ll} \operatorname{minimize} & c^{\mathrm{T}} x \ \text { subject to } & A x \quad=b \ & I x+I x^{\prime}=u \ & x \geq 0, x^{\prime} \geq 0 . \end{array}$$

## 数学代写|优化理论代写Optimization Theory代考|Problems with explicitly bounded variables

$$x_1+x_2+\cdots+x_n=1 \quad 0 \leq x_j \leq 1, \quad j=1, \ldots, n$$

$$x_1+x_2+\cdots+x_n=1 \quad x_j \geq 0, \quad j=1, \ldots, n$$

## 数学代写|优化理论代写Optimization Theory代考|Lower and upper bounds on variables

$$\text { minimize } \quad c^{\mathrm{T}} x \text { subject to } \quad A x=b, \quad l_j \leq x_j \leq u_j \quad \text { for } j=1,2, \ldots, n .$$

$$\text { minimize } \quad c^{\mathrm{T}} x \text { subject to } \quad A x \quad=b \quad I x+I x^{\prime}=u \quad x \geq 0, x^{\prime} \geq 0 .$$

## MATLAB代写

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