Not that you stated to the contrary, but the reality is that the theory and algorithms for convex optimization doesn't really differ that much from the theory o

Not that you stated to the contrary, but the reality is that the theory and algorithms for convex optimization doesn't really differ that much from the theory of general nonlinear optimization. Basically, we get different guarantees, but we're still largely applying the same algorithms (Newton based methods combined with things like projection, active set, or interior point methods for inequalities.) Though, I very strongly agree that knowing convex optimization tricks helps this process immensely. If you're going to formulate a problem, please, dear god, choose the convex one if given a choice. Mostly, this is a long way to say that if we're going to study optimization, we might as well study continuous optimization, which encompasses convex optimization and isn't that much more difficult. And, yes, this stuff is incredibly useful with applications in most fields of study. I've made a nice career out of it.