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# 数学代写|优化理论代写Optimization Theory代考|ELEN90026 Conversion to standard form

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## 数学代写|优化理论代写Optimization Theory代考|Conversion to standard form

A single linear inequality of the form
$$\sum_{j=1}^n a_{i j} x_j \leq b_i$$
can be converted to a linear equation by adding a nonnegative slack variable. Thus, the inequality above is equivalent to
$$\sum_{j=1}^n a_{i j} x_j+x_{n+i}=b_i, \quad x_{n+i} \geq 0 .$$
In this case, $x_{n+i}$ is the added slack variable.
Here is a simple illustration of this process. Suppose we have the linear inequality
$$3 x_1-5 x_2 \leq 15 .$$
At the moment, we are not imposing a nonnegativity constraint on the variables $x_1$ and $x_2$. This linear inequality can be converted into a linear equation by adding a nonnegative variable, $x_3$ to the left-hand side so as to obtain
$$3 x_1-5 x_2+x_3=15, \quad x_3 \geq 0 .$$
Figure $2.1$ depicts how the sign of the slack variable $x_3$ behaves with respect to the values of $x_1$ and $x_2$ in (2.2). The shaded region corresponds to the values of $x_1$ and $x_2$ satisfying the given linear inequality.

## 数学代写|优化理论代写Optimization Theory代考|Linear programs with free variables

As you will recall, a linear programming problem is an optimization problem in which a linear function is to be minimized (or maximized) subject to a system of linear constraints (equations or inequalities) on its variables. We insist that the system contain at least one linear inequality, but we impose no further conditions on the number of equations or inequalities. In particular, the variables of a linear program need not be nonnegative. Some linear programming problems have variables that are unrestricted in sign. Such variables are said to be free. Why do we care whether variables are free or not? This has to do with the fact that the Simplex Algorithm (which we shall take up in Chapter 3) is designed to solve linear programs in standard form.

In the following example, we encounter a classic optimization problem arising in statistics that does not appear to be a linear program. The prob-lem can, however, be converted to a linear program, albeit one that is definitely not in standard form. In particular, it will have linear inequality constraints and free variables. Later, we show how to bring this linear program into standard form.

## 数学代写|优化理论代写Optimization Theory代考|Conversion to standard form

$$\sum_{j=1}^n a_{i j} x_j \leq b_i$$

$$\sum_{j=1}^n a_{i j} x_j+x_{n+i}=b_i, \quad x_{n+i} \geq 0 .$$

$$3 x_1-5 x_2 \leq 15 .$$

$$3 x_1-5 x_2+x_3=15, \quad x_3 \geq 0 .$$

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