I want to know if it is still a convex function after function composition?

1

![58c8aaa7802a357eb0a2646fc088eca|690x216](upload://mGQxLBdeoQcvWIdvRxouGWT271u.png

58c8aaa7802a357eb0a2646fc088eca
58c8aaa7802a357eb0a2646fc088eca.png1035×325 63.6 KB

It has been proved that expression 1 is a convex function, where X1 and X2 are variables. When we replace X1 with the square of X1 and X2 with the square of X2, is the new expression convex?
thx

2

No, you can specialize formula (2) to produce expressions which look like -\log{(1+x^2)}, which is neither convex nor concave.

3

Thank you. It seems that the way I want to convert the problem to convex optimization is not feasible