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Converting a piecewise function to linear equations

Judging by the additional remarks in the comments, it seems like you want to linearize: $$ \alpha = \begin{cases} \alpha_1, \quad \text{if} \quad \sum_{i \in J_1} x_i = |J_1| = 1 \\ \alpha_2, \quad \...
joni's user avatar
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0 votes

Are there standard symbols to describe discrete event simulations (DES)

Kendall notation is okay up until a point for simple queueing systems. I have found that when I have a network of queues each having substantially different properties that the notation gets awkward. ...
Galen's user avatar
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Need help understanding robust optimization formulation

I believe there is a typo in the document, and perhaps some confusion between parameters $\text{u}$ and variables $u$. My understanding is that the initial problem $(P)$ is equivalent to: $$ \max_x \; ...
Kuifje's user avatar
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1 vote

Deploying optimization model to production environment

To validate your model, you need to make sure it captures the dynamics of your system correctly. To do so, you can constrain the model with your historical data (e.g. one month, or one year depending ...
Kuifje's user avatar
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3 votes

Lagrangian Relaxation Lower Bound exceeds the Upper bound and the Optimal solution

That sounds like a good choice for a relaxation, but you haven't explicitly shown how you compute the bounds. From your description so far, I suspect that you have two errors. First, the objective ...
RobPratt's user avatar
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1 vote

Can an Operations Research model optimize on a vector's indexes?

This is a complement to the answer given by @prubin. I will start by restating the definition you presented, just for conciseness. Let $E \in \mathbb{R}^{n^{n + 1}}$ be a "matrix"; $t \in \...
Matheus Diógenes Andrade's user avatar
3 votes
Accepted

How to model the following Constraint

As in How to enforce logical implication $\sum_j a_j x_j \le b \implies \sum_j c_j x_j \le d$, (temporarily) introduce constant $\epsilon>0$ and binary variable $Z$, and impose linear big-M ...
RobPratt's user avatar
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4 votes
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Can an Operations Research model optimize on a vector's indexes?

You can do this with ILP/MILP by using a binary variable $y_{i,j}$ indexed over $\lbrace 1,\dots, n\rbrace \times \lbrace 1,\dots, n\rbrace,$ where $y_{i,j}=1$ if and only if the $i$-th component of $...
prubin's user avatar
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1 vote

Formulation of a stepwise linear approximation

One approximation possible is to use the tangents of $f(x)$. Assuming you are minimizing $f(x)$, introduce a new variable $z$ to approximate $f(x)$ over a given set $A$ and use constraints: \begin{...
Kuifje's user avatar
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2 votes

Convex equivalent of a constraint

Introduce $d(e) \in \{0, 1\}$(or $[0, 1]$), $\forall e$. $$ \begin{aligned} -d(e) \leq a_1(e) - a_2(e) &\leq d(e)\\ \sum_e d(e) &\leq M \end{aligned} $$ Notice that we only need $d(e) \geq (...
xd y's user avatar
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3 votes
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Convex equivalent of a constraint

If $a_1$ and $a_2$ are binary, then $y=(a_1 - a_2)^2$ is also binary. More precisely: $$ a_1 \wedge a_2 \implies \neg y \quad \equiv \quad \neg(a_1 \wedge a_2) \vee \neg y \quad \equiv \quad \neg ...
Kuifje's user avatar
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