I recently watched a YouTube video titled “Stanford EE364/A Convex Optimization Stephen Boyd I 2023 I Lecture 4.” I’m having trouble understanding one of the examples Professor Boyd discussed regarding preserving convexity. Specifically, it pertains to the function z^2/y, which is a known convex function.

However, Professor Boyd mentioned that we can replace z with an affine function, for example, (1T*x), where x is a column vector. Additionally, we can replace y with min(2, sqrt(x^3)), which is a concave function. Professor Boyd explained that this substitution works because the function z^2/y is decreasing with respect to y, and min(2, sqrt(x^3)) is concave, satisfying the composition rule.

I have a doubt regarding this. The composition rule requires that z^2/y is non-increasing in each argument, so why can we ignore the monotonicity with respect to z just because (1T*x) is an affine function? It confuses me a lot.

I hope someone knowledgeable can provide a detailed explanation. Thank you very much!

I am unable to arrive at the same conclusion as the professor, regardless of the perspective from which I approach the preservation of convexity.