power¶

class cvxpy.power(x, p, max_denom=1024)[source]¶

Elementwise power function $f(x) = x^p$.

If expr is a CVXPY expression, then expr**p is equivalent to power(expr, p).

Specifically, the atom is given by the cases

\[\begin{split}\begin{array}{ccl} p = 0 & f(x) = 1 & \text{constant, positive} \\ p = 1 & f(x) = x & \text{affine, increasing, same sign as $x$} \\ p = 2,4,8,\ldots &f(x) = |x|^p & \text{convex, signed monotonicity, positive} \\ p < 0 & f(x) = \begin{cases} x^p & x > 0 \\ +\infty & x \leq 0 \end{cases} & \text{convex, decreasing, positive} \\ 0 < p < 1 & f(x) = \begin{cases} x^p & x \geq 0 \\ -\infty & x < 0 \end{cases} & \text{concave, increasing, positive} \\ p > 1,\ p \neq 2,4,8,\ldots & f(x) = \begin{cases} x^p & x \geq 0 \\ +\infty & x < 0 \end{cases} & \text{convex, increasing, positive}. \end{array}\end{split}\]

Note

Generally, p cannot be represented exactly, so a rational, i.e., fractional, approximation must be made.

Internally, power computes a rational approximation to p with a denominator up to max_denom. The resulting approximation can be found through the attribute power.p. The approximation error is given by the attribute power.approx_error. Increasing max_denom can give better approximations.

When p is an int or Fraction object, the approximation is usually exact.

Note

The final domain, sign, monotonicity, and curvature of the power atom are determined by the rational approximation to p, not the input parameter p.

For example,

>>> from cvxpy import Variable, power
>>> x = Variable()
>>> g = power(x, 1.001)
>>> g.p
Fraction(1001, 1000)
>>> g
Expression(CONVEX, POSITIVE, (1, 1))

results in a convex atom with implicit constraint $x \geq 0$, while

>>> g = power(x, 1.0001)
>>> g.p
1
>>> g
Expression(AFFINE, UNKNOWN, (1, 1))

results in an affine atom with no constraint on x.

When $p > 1$ and p is not a power of two, the monotonically increasing version of the function with full domain,

\[\begin{split}f(x) = \begin{cases} x^p & x \geq 0 \\ 0 & x < 0 \end{cases}\end{split}\]

can be formed with the composition power(pos(x), p).
The symmetric version with full domain,

\[f(x) = |x|^p\]

can be formed with the composition power(abs(x), p).

Parameters:

x : cvx.Variable

p : int, float, or Fraction

Scalar power.

max_denom : int

The maximum denominator considered in forming a rational approximation of p.