- #1

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Suppose we have [tex]m\times n[/tex] matrix [tex]\mathbf{A}[/tex], and [tex]n\times 1[/tex] column vector [tex]\mathbf{x}[/tex]. Then what do we mean by:

[tex]\mathbf{A}\mathbf{x}\leq \mathbf{b}[/tex]

geometrically?

Thanks in advance

- Thread starter EngWiPy
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- #1

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Suppose we have [tex]m\times n[/tex] matrix [tex]\mathbf{A}[/tex], and [tex]n\times 1[/tex] column vector [tex]\mathbf{x}[/tex]. Then what do we mean by:

[tex]\mathbf{A}\mathbf{x}\leq \mathbf{b}[/tex]

geometrically?

Thanks in advance

- #2

lurflurf

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- #3

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So, the intersection of all these planes is the polyhedron?

- #4

lurflurf

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Yes, in general they can be hyper planes.

- #5

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Ok thank youYes, in general they can be hyper planes.

- #6

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He says that, an efficient solution would be:

[tex]\text{max }\left(\prod_{i=1}^n\left(u_i-l_i\right)\right)^{1/n}[/tex]

[tex]\text{Subject to }\sum_{i=1}^n\left(a_{ij}^+u_j-a_{ij}^-l_j\right)\leq b_i,\,\,\text{ for }i=1,2,\ldots,n[/tex]

where [tex]a_{ij}^+=\text{max}\{a_{ij},0\}[/tex] and [tex]a_{ij}^-=\text{max}\{-a_{ij},0\}[/tex].

Here I have a couple of questions:

1- I assume that [tex]u_i-l_i[/tex] are the dimensions of the rectangle, and so, the volume will be the product of these dimensions, right? Then why we have the power of [tex]1/n[/tex] in the objective function?

2- How did he get the constraint? and why?

Any help in these two questions will be highly appreciated.

Thanks