p_norm {CVXR}R Documentation

P-Norm

Description

The vector p-norm. If given a matrix variable, p_norm will treat it as a vector and compute the p-norm of the concatenated columns.

Usage

p_norm(x, p = 2, axis = NA_real_, keepdims = FALSE, max_denom = 1024)

Arguments

x

An Expression, vector, or matrix.

p

A number greater than or equal to 1, or equal to positive infinity.

axis

(Optional) The dimension across which to apply the function: 1 indicates rows, 2 indicates columns, and NA indicates rows and columns. The default is NA.

keepdims

(Optional) Should dimensions be maintained when applying the atom along an axis? If FALSE, result will be collapsed into an n x 1 column vector. The default is FALSE.

max_denom

(Optional) The maximum denominator considered in forming a rational approximation for p. The default is 1024.

Details

For p ≥q 1, the p-norm is given by

\|x\|_p = ≤ft(∑_{i=1}^n |x_i|^p\right)^{1/p}

with domain x \in \mathbf{R}^n. For p < 1, p \neq 0, the p-norm is given by

\|x\|_p = ≤ft(∑_{i=1}^n x_i^p\right)^{1/p}

with domain x \in \mathbf{R}^n_+.

Value

An Expression representing the p-norm of the input.

Examples

x <- Variable(3)
prob <- Problem(Minimize(p_norm(x,2)))
result <- solve(prob)
result$value
result$getValue(x)

prob <- Problem(Minimize(p_norm(x,Inf)))
result <- solve(prob)
result$value
result$getValue(x)

## Not run: 
  a <- c(1.0, 2, 3)
  prob <- Problem(Minimize(p_norm(x,1.6)), list(t(x) %*% a >= 1))
  result <- solve(prob)
  result$value
  result$getValue(x)

  prob <- Problem(Minimize(sum(abs(x - a))), list(p_norm(x,-1) >= 0))
  result <- solve(prob)
  result$value
  result$getValue(x)

## End(Not run)

[Package CVXR version 1.0-10 Index]