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# 数学代写|线性规划代写Linear Programming代考|MTH503 Reusing a Factorization

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## 数学代写|线性规划代写Linear Programming代考|Reusing a Factorization

In the previous two sections, we showed how to use an $L U$-factorization of $B$ to solve the system of equations
$$B \Delta x_{\mathcal{B}}=a_j$$
for the primal step direction $\Delta x_{\mathcal{B}}$. Since the basis matrix doesn’t change much from one iteration of the simplex method to the next (columns get replaced by new ones one at a time), we ask whether the $L U$-factorization of $B$ from the current iteration might somehow be used again to solve the systems of equations that arise in the next iteration (or even the next several iterations).

Let $B$ denote the current basis (for which a factorization has already been computed) and let $\tilde{B}$ denote the basis of the next iteration. Then $\tilde{B}$ is simply $B$ with the column that holds the column vector $a_i$ associated with the leaving variable $x_i$ replaced by a new column vector $a_j$ associated with the entering variable $x_j$. This verbal description can be converted into a formula:
$$\tilde{B}=B+\left(a_j-a_i\right) e_i^T$$

The idea of writing the next basis as a product of the current basis times an easily invertible matrix can be extended over several iterations. For example, if we look $k$ iterations out, we can write
$$B_k=B_0 E_0 E_1 \cdots E_{k-1}$$
If we have an $L U$-factorization of $B_0$ and we have saved enough information to reconstruct each $E_j$, then we can use this product to solve systems of equations involving $B_k$.

Note that in order to reconstruct $E_j$, all we need to save is the primal step direction vector $\Delta x_{\mathcal{B}}^j$ (and an integer telling which column it goes in). In actual implementations, these vectors are stored in lists. For historical reasons, this list is called an eta-file (and the matrices $E_j$ are called eta matrices). Given the $L U$-factorization of $B_0$ and the eta-file, it is an easy matter to solve systems of equations involving either $B$ or $B^T$. However, as $k$ gets large, the amount of work required to go through the entire eta-file begins to dominate the amount of work that would be required to simply form a new $L U$-factorization of the current basis. Hence, the best strategy is to use an eta-file but with periodic refactorization of the basis (and accompanied purging of the eta-file).

## 数学代写|线性规划代写Linear Programming代考|Reusing a Factorization

$$B \Delta x_{\mathcal{B}}=a_j$$

$$\tilde{B}=B+\left(a_j-a_i\right) e_i^T$$

$$B_k=B_0 E_0 E_1 \cdots E_{k-1}$$

## MATLAB代写

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