p_norm function

P-Norm

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. methods

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.

Returns

An Expression representing the p-norm of the input.

Details

For $p \geq 1$, the p-norm is given by

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

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

• Note that the "p-norm" is actually a norm only when $p \geq 1$ or $p = +\infty$. For these cases, it is convex.
• The expression is undefined when $p = 0$.
• Otherwise, when $p \< 1$, the expression is concave, but not a true norm.

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)