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Is there a way to formulate the following Linear Program in a mixed-integer LP that I could solve with most linear programs in R/Python that support Mixed Integer Linear Programs (MILP)?

So my question is: How can I use a combination of integer, binary and continuous variables to reformulate the constraints (1) below?

Constants: $C_i$ (factor exposure), $x_i^a$ (initial weight)

Decision variables: $x_i$ (portfolio weight)

Portfolio Maximization:

$\max_{x_{i}}\sum_{i=1}^{N}x_{i}\cdot C_{i}$

subject to:

(1) $\boldsymbol{1}_{\left\{ x_{i}\geq x_{i}^{a}\right\} }\left(x_{i}-x_{i}^{a}\right)\in\{0\}\cup\left[0.025,\infty\right],\forall i$ (Minimum purchase size of 0.025)

where

$\boldsymbol{1}_{\left\{ x_{i}\geq x_{i}^{a}\right\} }=\begin{cases} 1 & \text{if } x_{i} \geq x_{i}^{a}\\ 0 & \text{otherwise} \end{cases}$

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  • $\begingroup$ did you already try something with binary variables? it seems rather standard $\endgroup$ – LinAlg Oct 25 '20 at 22:56
  • $\begingroup$ Hi, Yes I tried but I cannot figure out how to do it since there is a combination of semi-continous constraint (xi has to be 0 or above something), but this only for case when xi is bigger than xi^a => I only want to contraint the purchase, for a sale the ticket size doesn't matter $\endgroup$ – Adrien A. Oct 26 '20 at 2:17
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Introduce a small constant tolerance $\epsilon > 0$, binary decision variables $y_i^1$ and $y_i^2$, and linear constraints \begin{align} y_i^1 + y_i^2 &\le 1 &&\text{for all $i$} \tag1\\ (0-x_i^a) y_i^1 + 0.025 y_i^2 \le x_i - x_i^a &\le -\epsilon y_i^1 + (1-x_i^a)y_i^2 &&\text{for all $i$} \tag2\\ \end{align} Constraints $(1)$ and $(2)$ enforce $x_i - x_i^a \le -\epsilon \lor x_i - x_i^a = 0 \lor x_i - x_i^a \ge 0.025$. Explicitly, the three cases are \begin{align} (y_i^1,y_i^2)=(1,0): && -x_i^a \le x_i - x_i^a &\le -\epsilon \\ (y_i^1,y_i^2)=(0,0): && 0 \le x_i - x_i^a &\le 0 \\ (y_i^1,y_i^2)=(0,1): && 0.025 \le x_i - x_i^a &\le 1-x_i^a \\ \end{align}


More simply, introduce a binary decision variable $z_i$ and linear constraints $$-x_i^a (1-z_i) + 0.025 z_i \le x_i - x_i^a \le (1-x_i^a)z_i \text{ for all $i$}$$

If $z_i=0$, the constraint implies $-x_i^a \le x_i - x_i^a \le 0$, so $x_i \le x_i^a$.

If $z_i=1$, the constraint implies $0.025 \le x_i - x_i^a \le 1-x_i^a$, so $x_i \ge x_i^a + 0.025$.

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  • $\begingroup$ Thanks RobPratt! Actually I was more interested in how to solve for the first constraint in my problem. I edited the question where I only focus on (1). Do you know how would that work? $\endgroup$ – Adrien A. Oct 26 '20 at 2:12
  • $\begingroup$ I edited my answer to match your edited question. Just check the three cases for $(y_i^1,y_i^2)$ to see how it works. $\endgroup$ – RobPratt Oct 26 '20 at 2:18
  • $\begingroup$ Hi Rob, I tried but it doesn't seem to work. Maybe in (2) it should be (1-yi2)xia at the end? $\endgroup$ – Adrien A. Oct 26 '20 at 2:30
  • $\begingroup$ I edited my question and removed the 100 to make it simpler $\endgroup$ – Adrien A. Oct 26 '20 at 2:31
  • $\begingroup$ I added details of the three cases. What didn't work? $\endgroup$ – RobPratt Oct 26 '20 at 3:27

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