In a linear program, I would like some variables to: 1. Take the same values 2. Group some variables i.e. some variables should take same values or lie within certain percentage. 3. All different values
The objective function is given by:
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$\begingroup$ You specified $x_i \ge 0$ for all $i$. For (2), did you maybe mean $>0$ instead of $\ge 0$? $\endgroup$– RobPrattApr 30, 2020 at 0:53
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$\begingroup$ Yes, I intended to say greater than zero in (2) and not greater than equal to $\endgroup$– SamApr 30, 2020 at 9:08
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1 Answer
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(1) This is correct, and there's nothing wrong with having a whole lot of constraints that each require $x_6 = x_j$ for some $j$. But if you know in advance that these variables will all equal each other, why not just define a new variable that equals all of them? That is, create a variable $x_{6-10}$ that equals $x_6$ through $x_{10}$ and use this variable everywhere any of the $x_6$ through $x_{10}$ variables appear?
(2) Using the logic described here, you can create a binary variable $y_6$ that equals 1 if $x_6 \ge 0$, and another binary variable $y_8$ that equals 1 if $x_8 \ge 0$. Then you can enforce the "if-then" implications using the logic described here.
(3) Some solvers have a built-in feature that allows you to specify that certain variables must be different from each other; see Matrix in ampl: constraint that the values are all different. But I think this only works for integer variables. For continuous variables, I think you need to use big-Ms for this, although maybe others will chime in with better ideas.
answered Apr 30, 2020 at 1:00
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$\begingroup$ Thank you. As Rob pointed out the ambiguity in my question (2), I intended to say greater than zero in (2) and not greater than equal to zero. Would this change your answer? $\endgroup$– SamApr 30, 2020 at 9:09
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$\begingroup$ You can't really do strict inequalities for continuous variables. The best you can do is something like $x_i \ge \delta$ for some small $\delta > 0$. Then the logic described in the answers linked to above still applies. $\endgroup$ Apr 30, 2020 at 14:11