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2 votes
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MIP formulation for a lower semi-continuous function

You want to maximize $\max(f(x),0)$. Assume $L \le f(x) \le U$ for constants $L$ and $U$. Maximize $g(x)$ subject to $$ 0 \le g(x) \le U y \\ 0 \le g(x) - f(x) \le (0-L)(1-y) \\ y \in \{0,1\} $$
RobPratt's user avatar
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3 votes
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Linearize piecewise function without big-M constraints

I don't see or remember a way around the big-M constraints. Thankfully they are not that hard to write. Or you may use indicator constraints which are even simpler. We add new binary variables, and ...
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2 votes

Travelling salesman problem

The MTZ constraints are big-M constraints that enforce the logical implication $$x_{ij}=1\implies u_j\ge u_i+d_j.$$ That is, if you traverse arc $(i,j)$, the cumulative demand at $j$ is at least the ...
RobPratt's user avatar
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