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The extremes of an nth dimension linear equation

  1. Sep 30, 2015 #1
    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, xn], 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?
     
  2. jcsd
  3. Sep 30, 2015 #2

    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.
     
  4. Sep 30, 2015 #3

    andrewkirk

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    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.
     
  5. Oct 1, 2015 #4
    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.
     
    Last edited: Oct 1, 2015
  6. Oct 1, 2015 #5

    andrewkirk

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    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.
     
  7. Oct 1, 2015 #6
    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.
     
  8. Oct 4, 2015 #7

    Stephen Tashi

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    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.
     
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