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Thread: Maximize multiple linear equations simultaneously

  1. #1
    Mar 2011

    Maximize multiple linear equations simultaneously

    This is probably best placed in linear algebra, but I don't know exactly where this fits, possibly statistical analysis or business math or some sort. Anyway I have a matrix that looks something like this

    [ 1 1 0 5 1|A ]
    [ 0 3 0 1 1|B ]
    [ 4 1 1 0 1|C ]
    [ 1 0 3 1 0|D ]

    and the independant variables may be a, b, c, d, e if you will. What I am trying to do is maximize A, B, C, and D under the single constraint which may be a + b + c + d + e = x, x some arbitrary value assigned. I am not looking to maximize one single equation but all four equations simultaneously. I have done some searching for some kind of method to do this and have come up only with single linear equations being optimized. I do not pretend to be particularly handy with mathematics, or matrix math for that matter, but I am very interested if there is indeed a method to optimize multiple linear equations simultaneously under a single constraint. If not, I may have found a life goal!

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  2. #2
    A Plied Mathematician
    Jun 2010
    CT, USA
    I'm not an expert in optimization, but I see a problem with the way your problem is set up. You mention that most of the information you've found relates to optimizing single (linear, but it doesn't necessarily have to be linear) equations. There's a reason for that: you should generally have a "value" function, or a "feedback" function - a function that tells you how well you're doing in your optimization search. Is this solution better than that one or not? In your case, there's no clear-cut way to say that {A,B,C,D} = {1,0,0,0} is better or worse than {A,B,C,D} = {0,1,0,0}. This essentially comes down to the fact that the real numbers obey the law of trichotomy, and is the largest number system that does so, unless I'm mistaken. Certainly, vectors in $\displaystyle \mathbb{R}^{4},$ like {A,B,C,D}, are NOT ordered. Hence, you can't compare them with an equals sign, a less-than sign, or a greater-than sign.

    So the question I have for you is this: what is your value function? How are you to tell whether one solution is better than another or not?
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