The extremes of an nth dimension linear equation

[worryingchem]

Hi,
I wanted to know if the endpoints of an nth dimension linear equation will be guaranteed to contain a min and max over that interval.
For 1D ( like a line), if I find f(x) over an interval [x₀, x_n], I'm guaranteed that the two end points will be either an max or min.
So I was wondering if this applies to any nth dimension linear equation?

 

[Hornbein]

Hi,
I wanted to know if the endpoints of an nth dimension linear equation will be guaranteed to contain a min and max over that interval.
For 1D ( like a line), if I find f(x) over an interval [x₀, x_n], I'm guaranteed that the two end points will be either an max or min.
So I was wondering if this applies to any nth dimension linear equation?

It seems to me that this true pretty much by the definition of linearity. Indeed, you don't even need linearity or even continuity, the function being monotonic is enough. I don't see what the number of dimensions has to do with it. So maybe I don't really understand the question.

 

[andrewkirk]

If you define 'endpoints' broadly enough then yes. The endpoints in the n-dimensional case are the ##2^n## vertices of the n-dimensional polytope (generalisation of polygon) that is the constrained region.

 

[worryingchem]

Thank you for answering my question.
I wanted to know that if I solve the following system over an interval of b:

## \left[ \begin{array}{c} a \ b \\ c \ d \\ \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \ \end{array} \right] = \left[ \begin{array}{c} [b_{11}, b_{1n}] \\ [b_{21}, b_{2n}] \ \end{array} \right] ##

Then, any linear combinations of the x values ##( c_1x_1 + c_2x_2 )## at those 4 endpoints will contain the min and max. And that this will also apply to any nth dimension linear equations.

 

[andrewkirk]

I'm afraid I'm not sure what you mean.

Are you asking whether, given a n x n real matrix M, n real intervals ##[b_{i1},b_{i2}], 1\leq i\leq n##, and a linear function ##f:\mathbb{R}^n\to\mathbb{R}##, the maximum value of ##f## over the 'hyper-rectangular' set

$$S=\left\{\vec{x}\in\mathbb{R}^n\ \big| \ M\vec{x}\in\prod_{i=1}^n[b_{i1},b_{i2}]\right\}$$

occurs at one of the points
$$\big(b_{1k_1},b_{2k_2},...,b_{nk_n}\big)$$
where every ##k_j## is in {1,2} (And the minimum of ##f## over ##S## also occurs at one of those points)?

The answer to that is Yes.

 

[worryingchem]

Yes, I think what you said is what I wanted, if for every dimension, each ## b_{nk_n} ## is a pair of endpoints to the interval.

 

[Stephen Tashi]

Hi,
I wanted to know if the endpoints of an nth dimension linear equation will be guaranteed to contain a min and max over that interval.

As a technicality, you should ask about the max and min of a linear "function" instead of using the terminology linear "equation".

It's isn't clear what you mean by "that interval". I suppose you are thinking of a line segment on the graph of a linear equation in two variables, but linear equations in higher dimensions can describe more general geometric figures. For example, in 3D, the equation x + 2y = 3 doesn't put any constraints on the z value.