Disciplined convex programming representation of $x\sqrt{1-x}$

Question

Anyone have an idea for a disciplined convex programming (DCP) representation of the concave function $x\sqrt{1-x}$, which is defined over the domain $[0,1]$?

The Taylor series about $x=0$ is $$x - \frac{x^2}{2} - \frac{x^3}{8} - \frac{x^4}{16} + O(x^5),$$ which is DCP. However, I'm trying to find a DCP representation for the function itself, preferably.

Answer 1

I think in convex functions and epigraphs, so I hope you don't mind me deriving a DCP representation of the inequality $$t \geq -x \sqrt{1-x},$$

on $x \in (-∞, 1]$, which I rewrite to $$t \geq (1-x) \sqrt{1-x} - \sqrt{1-x} = r^{3/2} - r^{1/2},\quad r = 1-x \geq 0.$$

It should now be easy to input into your DCP framework. In MOSEK, it could be represented using two power cones: $$ t \geq p - q,\quad p \geq |r|^{3/2},\quad |q| \leq \sqrt{r}.$$

Answer 2

I don't think you can represent this as a concave function in the (cvxpy-)DCP sense:

import cvxpy as cp
x=cp.Variable()
a=x*cp.sqrt(1-x)
a.curvature

'UNKNOWN'

However, you can represent it using DQCP (disciplined quasiconvex programming):

import cvxpy as cp
x=cp.Variable(pos=True)
a=x*cp.sqrt(1-x)
a.curvature

'QUASICONCAVE'

Answer 3

This is possible purely under DCP. As you are interested in the interval $[0,1]$, rewrite your function as $$x\sqrt{1-x}=\exp\left(\ln x+\frac12\ln(1-x)\right),\quad x\in[0,1].$$ Then the following code

import cvxpy as cp
x = cp.Variable()
y = cp.exp(cp.log(x) + 0.5 * cp.log(1 - x))
print("x * sqrt(1 - x) curvature:", y.curvature)

gives QUASICONCAVE as desired.

Answer 4

Does this appear in a context in your program such that it can be replaced with a strictly monotonically increasing function of itself? For example, if it is the only term in the objective function, or the only term on the LHS of a LHS >= constant constraint.

If so, this can be handled as a weighted-geometric mean, which actually represents $(x\sqrt{1-x})^{2/3}$.

In CVX (the original DCP tool), this can be done using the geo_mean function, as follows:

cvx_begin
variable x
maximize(geo_mean([x;1-x],1,[2;1]))
<add any constraints>
cvx_end

Internally, this geo_mean formulation is handled by CVX in a manner similar to @TheSimpliFire 's answer.

help geo_mean
 geo_mean   Geometric mean.
    Y=geo_mean(X), where X is a vector, computes the geometrix mean of X. If any
    of the elements of X are negative, then Y=-Inf. Otherwise, it is equivalent
    to Y=PROD(X).^(1/LENGTH(X)). All elements must be real.
For matrices, geo_mean(X) is a row vector containing the geometric means of
the columns. For N-D arrays, geo_mean(X) is an array of the geometric means
taken along the first non-singleton dimension of X.

geo_mean(X,DIM) takes the geometric mean along the dimension DIM of X.

geo_mean(X,DIM,W), where W is a vector of nonnegative integers, computes a
weighted geometric mean Y = PROD(X.^W)^(1/SUM(W)). This is more efficient
than replicating the values of X W times. Note that W must be a vector,
even if X is a matrix, and its length must be the same as SIZE(X,DIM).

Disciplined convex programming information:
    geo_mean is concave  and nondecreasing; therefore, when used in CVX
    specifications, its argument must be concave.

EDIT: @Henrik Alsing Friberg's subsequent answer is better than mine. I showed how to represent the stated function to the 2/3 power (which would only be useful in the circumstances I described), whereas he showed how to represent the function as stated in the question.