How to Imagine This?

Hi,

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

$$\mathbf{A}\mathbf{x}\leq \mathbf{b}$$

geometrically?

Thanks in advance

Answers and Replies

lurflurf
Homework Helper
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.

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?

lurflurf
Homework Helper
Yes, in general they can be hyper planes.

Yes, in general they can be hyper planes.
Ok thank you

Dr. Boyd at Stanford University, says the following in the solution of a homework: The question is: find the maximum volume rectangle $$\mathbf{R}=\{\mathbf{x}:\mathbf{l}\leq\mathbf{x}\leq\mathbf{u}\}$$ in a polyhedron $$\mathbf{P}=\{\mathbf{x}:\mathbf{A}\mathbf{x}\leq\mathbf{b}\}$$.

He says that, an efficient solution would be:

$$\text{max }\left(\prod_{i=1}^n\left(u_i-l_i\right)\right)^{1/n}$$
$$\text{Subject to }\sum_{i=1}^n\left(a_{ij}^+u_j-a_{ij}^-l_j\right)\leq b_i,\,\,\text{ for }i=1,2,\ldots,n$$
where $$a_{ij}^+=\text{max}\{a_{ij},0\}$$ and $$a_{ij}^-=\text{max}\{-a_{ij},0\}$$.

Here I have a couple of questions:

1- I assume that $$u_i-l_i$$ 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 $$1/n$$ in the objective function?

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

Any help in these two questions will be highly appreciated.

Thanks