How to Imagine This?

  • Thread starter EngWiPy
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  • #1
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Hi,

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
 

Answers and Replies

  • #2
lurflurf
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It looks like the constraint system if a linear program. The system in most cases defines a polyhedral (that can be shown to be convex) that restricts x to its interior. This occurs in many areas such as sandwich making, in that case we can make n types of sandwiches, we are constained by m conditions on our sandwiches such as our available mustard or number of lemon peal sandwich rolls. We can visualize our posible set of sandwiches by a polyhedra.
 
  • #3
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It looks like the constraint system if a linear program. The system in most cases defines a polyhedral (that can be shown to be convex) that restricts x to its interior. This occurs in many areas such as sandwich making, in that case we can make n types of sandwiches, we are constained by m conditions on our sandwiches such as our available mustard or number of lemon peal sandwich rolls. We can visualize our posible set of sandwiches by a polyhedra.
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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Yes, in general they can be hyper planes.
Ok thank you
 
  • #6
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Dr. Boyd at Stanford University, says the following in the solution of a homework: The question is: find the maximum volume rectangle [tex]\mathbf{R}=\{\mathbf{x}:\mathbf{l}\leq\mathbf{x}\leq\mathbf{u}\}[/tex] in a polyhedron [tex]\mathbf{P}=\{\mathbf{x}:\mathbf{A}\mathbf{x}\leq\mathbf{b}\}[/tex].

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
 

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