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

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.

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: $$t \geq p - q,\quad p \geq |r|^{3/2},\quad |q| \leq \sqrt{r}.$$

• Nice, $t \ge r^{3/2} - r^{1/2}, r = 1-x \ge 0.$ should go right into CVX, CVXPY, and CVXR. This should be the answer accepted by the OP. – Mark L. Stone Jan 3 '20 at 13:33

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.

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'

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]))
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.

• Thanks Mark - This is very helpful. I should have been more explicit in my post. I am actually looking for a variable transformation that will allow me to represent $f = \sum(y_i^3)/\sum(y_i^2\sqrt{1-y_i^2})$ as a quasiconvex function. I need to preserve convexity in the numerator and concavity in the denominator. Your suggestion is spot on and I had not considered using geo_mean. Introducing the change of variables that $z_i = y_i^3$ presents an affine numerator and allows the denominator to be represented as sum(geo_mean([z_i;z_i^(1/3)-z_i])), which is concave. Very helpful and thanks. – Dan Berkenstock Dec 31 '19 at 4:20