How to linearize a $1$-norm equality constraint?

Asked Feb 14 '19 at 13:30

Active Aug 14 '21 at 10:15

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Let $x, y \in \mathbb{R}^n$ be fixed vectors of $1$-norm $C$. My optimization problem is the following

$$\begin{array}{ll} \underset{\beta \in \Bbb{R}^n}{\text{minimize}} & | \beta |_M \\ \text{subject to} & | x + \beta |_1 = C\\ & |x + \beta - y|_1 \leq D\end{array}$$

where $M \in \{ 1, 2, \infty \}$. I can choose it to be $M=2$ and my problem can easily become with quadratic target function. I Also can easily 'linearize' the $|x + \beta - y|_1 \leq D$ condition, but I find it hard to linearize $ |x+\beta|_1 = C$. One can make it $$ |x+\beta|_1 \leq C \\ |x+\beta|_1 \geq C $$ but again, I can't figure how to linearize the second one, $|x+\beta|_1 \geq C$.

edited Aug 14 '21 at 10:15

Rodrigo de Azevedo

asked Feb 14 '19 at 13:30

Veliko

1,084

1

This constraint is not convex. You need to add binary variables that can model a piecewise nonlinear function, see section 7.6 in this document – LinAlg Feb 14 '19 at 13:33
I made a little mistake. My vectors $x,y$ are of $l_1$ norm $C$. This would immediately yield the $\beta = 0^n$ for most cases for $C$ and $D$ and I don't want it. In my case I can't relax. – Veliko Feb 14 '19 at 14:04
Substitute non linear constraint for equivalent set of linear inequalities and/or equalities. – Veliko Feb 14 '19 at 14:25
Take a look at this answer. – Rodrigo de Azevedo Feb 14 '19 at 14:44

How to linearize a $1$-norm equality constraint?

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