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For a project I need something solved, it screams linear programming. If I get the problem in "standard" form I should be able to solve it using the simplex method. But I don't see how to get it in standard form. The problem is this:

Maximize $$min(f_1(\lambda),...,f_p(\lambda))$$ s.t $\sum \lambda_i=1,\lambda_i \geq 0$

where $\lambda=(\lambda_1,...,\lambda_n)$ and $f_i$ linear.

Help or direction to a good source would be much appreciated!

user153227
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1 Answers1

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This is a basic case of optimiation problem easy to be cast as a linear one, you can find it in any textbook.

$\max t$

$ t\leq f_i(\lambda),\quad i=1...$

$\sum \lambda_i = 1$

$\lambda_i\geq 0, \quad i=1...$

Richard
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AndreaCassioli
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    This is now the top result for the Google search 'maximize minimum of linear functions'. It's feels a bit condescending to read that one should search carefully before trying to find an answer in SE. – JiK Jun 14 '17 at 05:36
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    Well, when I wrote the answer it was not....I found things like http://www.4er.org/CourseNotes/Book%20A/A-III.pdf – AndreaCassioli Jun 14 '17 at 06:39
  • How can it be solved? By simplex method? if so, is there any demonstration? Also, could someone provide some references about the computational complexity of this problem? – jerry Dec 27 '23 at 14:54
  • Any linear programming solver would do (it will use either simplex or interior point algorithm under the hood). As for complexity, linear programming is broadly speaking polynomially solvable (https://en.wikipedia.org/wiki/Linear_programming#Algorithms). – AndreaCassioli Jan 03 '24 at 09:25
  • I found that the proof of the time complexity in this maximin problem is missing. Are there any references? Thank you! – jerry Feb 15 '24 at 08:03