Overview
One of the key questions when optimizing a function is whether the function is convex to begin with. This question can often be effectively answered simply by understanding the operations that preserve convexity and showing how the function in question can be composed from this list.
Operations
- Multiplication by a Constant - If {% f(x) %} is convex, then {% af(x) %} is convex
- Addition - if {% f_1(x) %} and {% f_2(x) %} are both convex, then {% f_1(x) + f_2(x) %} is convex
- Affine Mapping - if {% f(\vec{x}) %} is convex, then {% f(A\vec{x} + \vec{b}) %} is convex.
- Pointwise Maximum and Minimum - if {% f_1(x) %} and {% f_2(x) %} are both convex, then {% max(f_1(x),f_2(x)) %} is convex