How to solve the constraint "sqrt(x)(1+e^(-x)) >=c", where x >= 0，and c is a constant parameter

Nonconvex
1

Here, f(x) = sqrt(x)(1+e^(-x)), and f’(x) >= 0. But f(x) is non-convex and non-concave. How to transform this constraint approximately and solve it by CVX?

2

The LHS needs to be concave, but is neither concave nor convex.

2nd derivative at x = 2 is negative. 2nd derivative at x= 2.5 is positive.

3

Thanks for your kind reply. I understand that this constraint is non-convex and non-concave, so it needs another transformation trick to solve it.

5

if f(x) is monotonically increasing, then fx>=c can just be x>=d, you need to find the constant d.

6

That’s a great idea. f(x) = sqrt(x)(1+e^(-x)) is monotonically increasing with respect to x.