Matlab: how do I run the optimization (fmincon) repeately? - optimization

Question

I am trying to follow the tutorial of using the optimization tool box in MATLAB. Specifically, I have a function
f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b
subject to the constraint:
(x(1))^2+x(2)-1=0,
-x(1)*x(2)-10<=0.
and I want to minimize this function for a range of b=[0,20]. (That is, I want to minimize this function for b=0, b=1,b=2 ... and so on).
Below is the steps taken from the MATLAB's tutorial webpage(http://www.mathworks.com/help/optim/ug/nonlinear-equality-and-inequality-constraints.html), how should I change the code so that, the optimization will run for 20 times, and save the optimal values for each b?
Step 1: Write a file objfun.m.
function f = objfun(x)
f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b;
Step 2: Write a file confuneq.m for the nonlinear constraints.
function [c, ceq] = confuneq(x)
% Nonlinear inequality constraints
c = -x(1)*x(2) - 10;
% Nonlinear equality constraints
ceq = x(1)^2 + x(2) - 1;
Step 3: Invoke constrained optimization routine.
x0 = [-1,1]; % Make a starting guess at the solution
options = optimoptions(#fmincon,'Algorithm','sqp');
[x,fval] = fmincon(#objfun,x0,[],[],[],[],[],[],...
#confuneq,options);
After 21 function evaluations, the solution produced is
x, fval
x =
-0.7529 0.4332
fval =
1.5093
Update:
I tried your answer, but I am encountering problem with your step 2. Bascially, I just fill the my step 2 to your step 2 (below the comment "optimization just like before").
%initialize list of targets
b = 0:1:20;
%preallocate/initialize result vectors using zeros (increases speed)
opt_x = zeros(length(b));
opt_fval = zeros(length(b));
>> for idx = 1, length(b)
objfun = #(x)objfun_builder(x,b)
%optimization just like before
x0 = [-1,1]; % Make a starting guess at the solution
options = optimoptions(#fmincon,'Algorithm','sqp');
[x,fval] = fmincon(#objfun,x0,[],[],[],[],[],[],...
#confuneq,options);
%end the stuff I fill in
opt_x(idx) = x
opt_fval(idx) = fval
end
However, it gave me the output is:
Error: "objfun" was previously used as a variable, conflicting
with its use here as the name of a function or command.
See "How MATLAB Recognizes Command Syntax" in the MATLAB
documentation for details.

Answer 1

There are two things you need to change about your code:
Creation of the objective function.
Multiple optimizations using a loop.
1st Step
For more flexibility with regard to b, you need to set up another function that returns a handle to the desired objective function, e.g.
function h = objfun_builder(x, b)
h = #(x)(objfun(x));
function f = objfun(x)
f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1) + b;
end
end
A more elegant and shorter approach are anonymous functions, e.g.
objfun_builder = #(x,b)(exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1) + b);
After all, this works out to be the same as above. It might be less intuitive for a Matlab-beginner, though.
2nd Step
Instead of placing an .m-file objfun.m in your path, you will need to call
objfun = #(x)(objfun_builder(x,myB));
to create an objective function in your workspace. In order to loop over the interval b=[0,20], use the following loop
%initialize list of targets
b = 0:1:20;
%preallocate/initialize result vectors using zeros (increases speed)
opt_x = zeros(length(b))
opt_fval = zeros(length(b))
%start optimization of list of targets (`b`s)
for idx = 1, length(b)
objfun = #(x)objfun_builder(x,b)
%optimization just like before
opt_x(idx) = x
opt_fval(idx) = fval
end