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.

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.

Value

An Expression representing the p-norm of the input.

Details

For $$p \geq 1$$, the p-norm is given by $$\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$$ with domain $$x \in \mathbf{R}^n$$. For $$p < 1, p \neq 0$$, the p-norm is given by $$\|x\|_p = \left(\sum_{i=1}^n x_i^p\right)^{1/p}$$ 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$value #> [1] 0 result$getValue(x)
#>      [,1]
#> [1,]    0
#> [2,]    0
#> [3,]    0

prob <- Problem(Minimize(p_norm(x,Inf)))
result <- solve(prob)
result$value #> [1] -2.775514e-23 result$getValue(x)
#>               [,1]
#> [1,] -3.231174e-27
#> [2,]  0.000000e+00
#> [3,]  0.000000e+00

if (FALSE) {
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)
}