I don’t believe that either of these points are true. In your original example, there is one correct solution for any convex function. I will assume there is a single hard-coded function for the following, but it can be extended to work for an arbitrary function.
The output register having the correct solution is the target set.
The output register having any state is the basin of attraction.
Clearly any specific number (or rather singleton of that number) is a subset of all numbers, so the target is a subset of the basin. And further, because “all numbers” has more than one element, the target set is smaller than the basin.
Pretty much, yes, according to definition given. Like I said, not a particularly interesting optimization but an optimization none the less.
To extend on this, the basin of optimization is not any smaller than an iterative process acting on a single register (and if you loop the program, then the time horizon is the same). In both cases your basin is anything in that register and the target state is one particular number in that register. As far as I can tell the definition doesn’t have any way of saying that one is “more of an optimizer” than the other. If anything, the fixed output is more optimized because it arrives more quickly.
Ok, well, it seems like the one-shot non-iterative optimizer is an optimizer in a MUCH stronger sense than a random program, and I’d still expect a definition of optimization to say something about the sense in which that holds.
I don’t believe that either of these points are true. In your original example, there is one correct solution for any convex function. I will assume there is a single hard-coded function for the following, but it can be extended to work for an arbitrary function.
The output register having the correct solution is the target set.
The output register having any state is the basin of attraction.
Clearly any specific number (or rather singleton of that number) is a subset of all numbers, so the target is a subset of the basin. And further, because “all numbers” has more than one element, the target set is smaller than the basin.
This argument applies to literally any deterministic program with nonempty output. Are you saying that every program is an optimizer?
Pretty much, yes, according to definition given. Like I said, not a particularly interesting optimization but an optimization none the less.
To extend on this, the basin of optimization is not any smaller than an iterative process acting on a single register (and if you loop the program, then the time horizon is the same). In both cases your basin is anything in that register and the target state is one particular number in that register. As far as I can tell the definition doesn’t have any way of saying that one is “more of an optimizer” than the other. If anything, the fixed output is more optimized because it arrives more quickly.
Ok, well, it seems like the one-shot non-iterative optimizer is an optimizer in a MUCH stronger sense than a random program, and I’d still expect a definition of optimization to say something about the sense in which that holds.