Questions tagged [absolute-value]

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

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linearizing a constraint involving an absolute function

I would like to know what is the best way to linearize a constraint involving an absolute function. More precisely, imagine I have three binary variables and their relationships is as follows: |x-y| = ...
1 vote
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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 ...
193 views

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) ...
343 views

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 ...
168 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 ...