Is there a proof that simultaneous optimization delivers better results then sequential optimization for non-separable, multivariable functions?

Question

$f(x,y)$ is a non-separable, non-negative real-valued function and $f(x,y)$ should be maximized over $x$ and $y$.

Is there a proof that the simultaneous maximization $$\max_{x,\ y} f(x,y)$$ delivers better results then the sequential maximization? $$\max_{x} \max_{y} f(x,y)$$

If $f(x,y)$ is separable, then the simultaneous and the sequential maximization are equal. If $f(x,y)$ is not separable then only the simultaneous optimization should convergence to the unique maximum, if exists.

Is there any proof for this?

Does this answer your question? Sequential maximization $\max_{x} \max_{y} f(x,y)$ vs. simultaneous maximization $\max_{x,y} f(x,y)$ — Randall, Oct 30 '20 at 17:10
@Randall i have seen this thread before, but it doesn't answer my question. The suggestion there is that the sequential optimization delivers better results than the simultaneous approach. I think it has to be the other way around since a group of features $x0$ and $y0$ may lead to the best classification results although each individual feature may not be the best one. — sebgrocp, Oct 30 '20 at 19:58
@sebgrocp - you appear to have overlooked the final phrase in the answer to that thread: "and the inequality in the other direction is obvious." I.e., Yanko is proving that sequential and simultaneous optimization are equal. — Paul Sinclair, Oct 31 '20 at 01:58

Is there a proof that simultaneous optimization delivers better results then sequential optimization for non-separable, multivariable functions?

0 Answers0