i have this constraint right here, which is not linear. How would i linearize such a product. $number_t$ is a positive integer and $new_t$ and $reset_t$ are binary.
$$number_t = (number_{t-1}+new_t)\cdot (1-reset_t)$$
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2https://or.stackexchange.com/questions/2903/how-to-linearize-the-multiplication-of-an-integer-and-a-binary-integer-variable?rq=1 – Kuifje Feb 13 '24 at 13:02
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2https://or.stackexchange.com/questions/37/how-to-linearize-the-product-of-two-binary-variables?rq=1 – Kuifje Feb 13 '24 at 13:02
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2Does this answer your question? How to linearize the multiplication of an integer and a binary integer variable? – SecretAgentMan Feb 13 '24 at 13:10
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1Does this answer your question? How to linearize the product of a binary and a non-negative continuous variable? – Rob Feb 21 '24 at 10:49
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Some comments already hinted at questions that give you the answer. In your specific example, this translates to the following:
\begin{align} 0 \leq number_t \leq (number_{t−1}+new_t) \newline number_t \geq (number_{t−1}+new_t)-reset_tM \newline number_t \leq M(1−reset_t) \end{align}
If $reset_t = 0$, through the first and second constraint we force $number_t$ to take value $number_{t-1}+new_t$. If $reset_t=1$, through the first and third constraint, we force $number_t$ to be 0. Where $M$ is a large enough number.
PeterD
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Thank you @PeterD. Assuming i also have $number_t = (number_{t-1}+new_t)\cdot (1-reset_t)-c_t + b_t$, which of both are also binary, how would i need to modify the constraints then? Thanks in advance? – Uni ewr Feb 13 '24 at 14:38
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1Would it work if, for example, I replace $number_t$ with $nr_t$ in your solution and then calculate $number_t=nr_t-c_t+b_t$? – Uni ewr Feb 13 '24 at 15:17
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