I think it's actually a hard problem to find the global minimum of this function. I wonder if you have tried any other optimizers to see how they compare. I tried random search and gridsearch (sampling without replacement on a grid), for 100 iterations, and bayesopt did a lot better than those.
There are a few things you can do to improve bayesopt in this case.
(1) Tell bayesopt that the function is deterministic by passing 'IsObjectiveDeterministic',true.
(2) Run it for more evaluations, e.g., 'MaxObjectiveEvaluations',100.
(3) Compress your objective function so it doesn't span many orders of magnitude by passing it through log1p. See modified code below.
When I did these 3 things, it did somewhat better and occasionally found the 0 point.
I suspect that the 0 point is in a hole without enough of a clue around it for the GP model to latch on to. I'm curious to hear how other optimizers perform on this problem.
clc
N=300;
% - let's create 300 points with gaussian shape
x = linspace(0,10,N)';
b = [1;3;2;2]; % model parameters
mf=@modelfun
y = mf(b,x);
% - let's create search parameters
num(1) = optimizableVariable('base', [0,9],'Type','integer');
num(2) = optimizableVariable('height', [0,9],'Type','integer');
num(3) = optimizableVariable('location',[0,9],'Type','integer');
num(4) = optimizableVariable('width', [0,9],'Type','integer');
% - define loss function
bf=@(num)bayesfun(num,x,y);
% - call bayesopt with initial parameters close to model values
results = bayesopt(bf,num, 'maxobj',100, 'isobj',true);
b0=table2array(results.bestPoint);
results.bestPoint
original_objective = exp(bayesfun(results.bestPoint,x,y)) - 1 % Undo log1p
% let's see plot of model data and fit data
%------------------------
plot(x,y,'.b')
hold on
plot(x,mf(b0,x),'.r')
hold off
legend('y','fit')
function out=bayesfun(t,x,y)
y1=modelfun(table2array(t),x);
out=sum((y1-y).^2);
out = log1p(out);
end
function out=modelfun(b,x)
out = b(1);
out= out + b(2)*exp(-((b(3)-x)/b(4)).^2);
end
