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.
p_norm(x, p = 2, axis = NA_real_, keepdims = FALSE, max_denom = 1024)
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: |
keepdims |
(Optional) Should dimensions be maintained when applying the atom along an axis? If |
max_denom |
(Optional) The maximum denominator considered in forming a rational approximation for p. The default is 1024. |
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_+.
Note that the "p-norm" is actually a norm only when p ≥q 1 or p = +∞. 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.
An Expression representing the p-norm of the input.
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)
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