Function Repository Resource:

# LInfinitySolve

Solve the linear minimax problem

Contributed by: Jan Mangaldan
 ResourceFunction["LInfinitySolve"][m,b] finds an x that solves the linear minimax problem for the matrix equation m.x==b.

## Details

The linear minimax problem is also referred to as linear norm minimization or Chebyshev norm minimization.
ResourceFunction["LInfinitySolve"][m,b] gives a vector x that minimizes Norm[m.x-b,].
All entries in the matrix m and the vector b must be real numbers.
SparseArray objects can be used in ResourceFunction["LInfinitySolve"].
ResourceFunction["LInfinitySolve"] takes the same options as LinearProgramming.

## Examples

### Basic Examples (1)

Solve a simple minimax problem:

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### Scope (2)

Create a 4×3 matrix, and b is a length-4 vector:

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Use exact arithmetic to find a vector x that minimizes :

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Use machine arithmetic:

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Use 20-digit-precision arithmetic:

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Use a sparse matrix:

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### Applications (4)

Here is some data:

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Find the line that best fits the data in the minimax sense:

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Find the quadratic that best fits the data in the minimax sense:

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Show the data with the two curves:

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### Properties and Relations (2)

For a vector b, LInfinitySolve is equivalent to ArgMin[Norm[m.x-b,∞],x]:

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Create a 5×2 matrix, and b is a length-5 vector:

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Solve the minimax problem:

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This is the minimizer of :

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It also gives the coefficients for the line with minimax absolute deviation from the points:

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## Version History

• 1.0.0 – 06 January 2021