performance tuning - Parallel function evaluation for minimal value

I am asking myself, if the evaluation of a function f looking for a minimal value over a discrete set of values can be parallelize (brute scan for a potential minimal value over a rectangular region with specified resolution). Is it possible or just not suitable for parallel computing do to the need to always compare to a reference which needs to happen in the main kernel? As far as I understood it in

Why won't Parallelize speed up my code?

the forced evaluation in the main kernel with SetSharedVariable can cause a significant lost in speed, which I think is the case in my horribly parallelized evaluation (see below). Any suggestions? I am pretty sure, I am just not seeing the obvious perspective. I dont want to use NMinimize, I only want to scan rapidly (if possible also in parallel) a rectangular region with specified resolution and pick up the minimal value. Sorry, if this is a duplicate, I was not able to find an answer. Thanks.

Minimal example:

Function:

f = Sin[x - z + Pi/4] + (y - 2)^2 + 13;

Sequential evaluation with do:

Clear[fmin]
fmin
n = 10^1*2;
fmin = f /. {x -> 0, y -> 0, z -> 0};
fmin // N
start = DateString[]
Do[
 ftemp = f /. {x -> xp, y -> yp, z -> zp};
 If[ftemp < fmin, fmin = ftemp];

 , {xp, 0, Pi, Pi/n}
 , {yp, -2, 4, 6/n}
 , {zp, -Pi, Pi, 2*Pi/n}
 ]
end = DateString[]
DateDifference[start, end, {"Minute", "Second"}]
fmin // N

Horribly parallelized evaluation

Clear[fmin]

fmin
n = 10^1*2;
fmin = f /. {x -> 0, y -> 0, z -> 0};
fmin // N
SetSharedVariable[fmin];
start = DateString[]
ParallelDo[
 ftemp = f /. {x -> xp, y -> yp, z -> zp};
 If[ftemp < fmin, fmin = ftemp];
 , {xp, 0, Pi, Pi/n}

 , {yp, -2, 4, 6/n}
 , {zp, -Pi, Pi, 2*Pi/n}
 ]
end = DateString[]
DateDifference[start, end, {"Minute", "Second"}]
fmin // N

Answer

Don't compare to a single (shared) main-kernel variable (fmin) on each kernel. Instead, allow each kernel to find the smallest of the points it has checked. Let each kernel have its own private fmin. Then you'll have $KernelCount candidates for the minimum. Finally select the smallest of these.

ParallelCombine is made for precisely this type of approach. It may be a good idea to use Method -> "CoarsestGrained".