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# 数学代写|优化理论代写Optimization Theory代考|Deleting an inequality or equality constraint

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## 数学代写|优化理论代写Optimization Theory代考|Deleting an inequality or equality constraint

The approach for deleting a constraint once an optimal solution has been obtained is the same whether it be an inequality or an equality. Accordingly, we will assume that we are deleting an equality constraint.

Suppose that after obtaining an optimal solution, say $\bar{x}$, we decide that a particular constraint $A_k \cdot x=b_k$ needs to be deleted. Even though $\bar{x}$ will be feasible for the modified problem, the deletion of this constraint will reduce the size of the basis and might result in dual infeasibility. To “physically” delete a constraint from the system and update the basis inverse (or factorization) could require significant computational work. It is simpler instead to make the constraint nonbinding, by which we mean that the lefthand side is no longer required to take on the value $b_k$.

We can make the $k$-th constraint nonbinding by introducing the free variable $x_{n+1}$ (in the $k$-th constraint only) and setting its cost coefficient to zero in the objective. That is,
$A_k \cdot x+x_{n+1}=b_k, \quad x_{n+1}$ is free.
(Note: We could consider setting the right-hand side to zero; however, a constant in a nonbinding constraint does not affect the optimal objective value, so we do not set $b_k=0$.) Next we can convert the revised constraint to fit into the standard form with the rest of the constraints by expressing $x_{n+1}$ as the difference of two nonnegative variables, that is
$$A_k \cdot x+x_{n+1}^{+}-x_{n+1}^{-}=b_k, \quad x_{n+1}^{+} \geq 0, x_{n+1}^{-} \geq 0 .$$
After this, we apply $B^{-1}$ to the two new columns, and price them out. One of them will price out positive, the other one negative, or both may price out zero. We pivot to bring the positive (or zero) reduced-cost column into the basis and continue, if necessary, until the algorithm terminats with an optimal solution or an indication that none exists.

## 数学代写|优化理论代写Optimization Theory代考|Ranging (Sensitivity analysis)

As mentioned earlier, ranging, or sensitivity analysis, has to do with the impact of small changes to the optimal solution. With ranging we examine the extent to which changes in a single element do not affect the optimal basis. For example, we would be interested in how much a right-hand side element $b_i$ could change without causing the optimal basis to change. This would be important for key resources where we would be concerned about the impact of market variations, or simply concerned that we were off in our estimate of its value. In a similar vein, it would be comforting to know that minor variations in cost (or elements of the data matrix) do not impact our solution much. Such analysis has its limitations since we are only discussing modifications of single elements; later, in Section 6.3, we shall consider changing multiple elements through the use of parametric programming.

Let $B=\left[A \cdot j_1 \cdots A \bullet j_m\right]$ be an optimal basis for the LP
$$\text { minimize } c^{\mathrm{T}} x \quad \text { subject to } A x=b, x \geq 0$$
where $A$ is $m \times n$. Let $I_{\bullet} k$ denote the $k$-th column of the $m \times m$ identity matrix. Changing only the $k$-th component of $b$, from $b_k$ to $\hat{b}k=b_k+\delta$, amounts to creating a new right-hand side vector, say $$\hat{b}=b+\delta I{\bullet} k$$

## 数学代写|优化理论代写Optimization Theory代考|Deleting an inequality or equality constraint

(注意: 我们可以考虑将右侧设置为零；但是，非约束约束中的常数不会影响最优目标值，因此我们] 不设置 $b_k=0$.) 接下来，我们可以将修改后的约束转换为与其余约束一起符合标准形式的表达式 $x_{n+1}$ 作为两个非负变量的差值，即
$$A_k \cdot x+x_{n+1}^{+}-x_{n+1}^{-}=b_k, \quad x_{n+1}^{+} \geq 0, x_{n+1}^{-} \geq 0$$

## 数学代写|优化理论代写Optimization Theory代考|Ranging (Sensitivity analysis)

$$\operatorname{minimize} c^{\mathrm{T}} x \quad \text { subject to } A x=b, x \geq 0$$

$$\hat{b}=b+\delta I \bullet k$$

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