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For a series of variables, I have got the constrain functions: $ax_{1}+bx_{2}+cx_{3}+dx_{4}=0$ $0 \le x_{1}, x_{2}, x_{3}, x_{4} \le 1$

Now I want to find the solution that the minimum value of $x_{1}, x_{2}, x_{3}, x_{4}$ is the greatest number in all solutions. For example, I have the following 4 solutions:

$$ \begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \\ 0.3 & 0.03 & 0.03 & 0.3 \\ 0.35 & 0 & 0 & 0.35 \\ 0.2 & 1 & 1 & 0.2 \\ 0.9 & 1 & 1 & 0 \\ \end{matrix} $$

In all the above solutions, the third one is what I want to find. So how can I compile the objective function for such optimization problem? Thanks!

ZYSTEEL
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  • The word "compile" here in connection with "the objective function" is not a familiar verb. Do you mean, how to express the objective? You've described the objective as maximizing the minimum value of $x_1,x_2,x_3,x_4$, and so the expression is rather straightforward if you introduce a fifth variable to be an upper bound on these first four unknowns. – hardmath Sep 08 '16 at 03:58

1 Answers1

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Add a new variable $y$ and constraints $y \leq x_1,\ldots,x_4$, and maximize $y$.

Yuval Filmus
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