# Questions tagged [absolute-value]

For questions related to the absolute value function, particularly as it is modeled in optimization problems.

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### transform minimize weighted sum of absolute value into a linear optimization

For example, we have an optimization problem $$\min \sum_{i=1}^{n} |w_{i} - a_{i}| b_{i} \quad \text{s.t.} \quad \sum_{i=1}^{n} c_i w_i = 0$$ and $a_i, b_i, c_i$ are given. How to convert it into a ...
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### Multiple absolute values with multiple variables in an LP

Assume that we have a LP with the constraint $$\sum_{j} \left(c_j x_j + |c_j x_j - \alpha_j + \beta_j|\right) \leq y$$ and $$\alpha_j + \beta_j \leq \lambda_j$$ for all $j$, where the (positive) ...
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### Absolute value in an equality constraint

What is the best way to model or represent an equality constraint which includes an absolute term in the expression: $$x = |y|$$ $x \in \mathbb{R^+}$ and $y \in \mathbb{R}$
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### How to write constraint with sum of absolutes in Integer Programming?

I found a solution for just one term here How can we formulate constraints of the form $$\sum_{i=1}^n |x_i -a_i| \ge K$$ in Mixed Integer Linear Programming ?
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### How to minimize the sum of absolute values

How can I solve a problem such as the following: $$\text{minimize}~~~ \sum_{i=1}^n |x_i| \\ \text{subject to}~~~ A x \geq b$$ ? Without the absolute values on the variables, it is a simple linear ...
161 views

### Linearizing objective function with absolute differences

I want to turn this objective function $$\max \sum_{i=1}^{N-1} \sum_{j=i+1}^N |TX_i^T - TX_j^T|$$ where $T$ is just a vector with increasing integers (e.g $[1 \ 2]$) and $X_i$ is a vector ...
I want to solve the following optimization problem $$\begin{array}{ll} \text{minimize} & | c^\top x |\\ \text{subject to} & A x \leq b\end{array}$$ Without the absolute value, this a ...