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has this example matrix A some special propertries, which might be useful? $$ \left[\begin{array}{rrrrrr} 3 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 4 & 3 & 0 & 0 &-1 & 2 & 0 & 0 \\ 5 & 4 & 3 & 0 &-2 &-1 & 2 & 0 \\ 0 & 5 & 4 & 3 & 0 &-2 &-1 & 2 \\ 0 & 0 & 5 & 4 & 0 & 0 &-2 &-1 \\ 0 & 0 & 0 & 5 & 0 & 0 & 0 &-2 \end{array}\right] $$

I want to find $$\min \|Ax\|_\max$$ where x is a binary vector and for each Block [here column 1-4 and 5-8 are two blocks] only one $x_i$ is 1. Each Block has the property that each element is shifted by one index.

I copied the matrix from here: Minimize $\|Ax-b\|$ where $x$ is a binary vector Its a slighty different objective but the matrix is basically the same. In the comments someone say something about block matrices and the kernel which i don't fully understand. Maybe someone could explain that a litte bit more?´

Thank you

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DerrenJackson
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1 Answers1

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This is easily cast as a mixed-integer linear program. Adding variables $z\in\mathbb{R}$ and $y\in\mathbb{R}^6$, we have $$\begin{array}{ll} \text{minimize} & z \\ \text{subject to} & y_i \leq z, ~i=1,2,...6 \\ & A x = y \\ & x_1 + x_2 + x_3 + x_4 = 1 \\ & x_5 + x_6 + x_7 + x_8 = 1 \\ & x_i \in \{0,1\}, ~ i=1,2,\dots, 8 \end{array}$$ Any modeling framework with MILP support should have no difficulty handling this problem.

If instead you intend to minimize the maximum element in absolute value, you need to do this: $$\begin{array}{ll} \text{minimize} & z \\ \text{subject to} & -z \leq y_i \leq z, ~i=1,2,...6 \\ & A x = y \\ & x_1 + x_2 + x_3 + x_4 = 1 \\ & x_5 + x_6 + x_7 + x_8 = 1 \\ & x_i \in \{0,1\}, ~ i=1,2,\dots, 8 \end{array}$$

But to be honest, I can see the solution to both problems by inspection: the optimal value is $3$, and any of the the following $x$ vectors will achieve it: $$(1,0,0,0,1,0,0,0),(0,1,0,0,0,1,0,0),(0,0,1,0,0,0,1,0),(0,0,0,1,0,0,0,1)$$

Michael Grant
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