Optimization. Convex nonlinear function. Exponential mixture

Dear comrades, I am trying to solve the problem on resolving the mixture of two exponentials. The data comes in the Y(response measured from the system) over t (time) histogram. The task I have formulated sounds like this: Minimize |yhat(c,t)-y|; where yhat = c(3)*exp(-t*c(1)) + c(4)*exp(-t*c(2)) subject to c(1-4)>=1 and c(3)+c(4)=1.

Here are my attempts in solving it (2 in total):

(1)With the use of Optimization Toolbox:
close all, clear all
t = [0 .3 .8 1.1 1.6 2.3]';
y = [.82 .72 .63 .60 .55 .50]';
plot(t,y,'o')
yhat = @(c,t) c(3)*exp(-t*c(1)) + c(4)*exp(-t*c(2)); 
c0=[0.5; 0.5; 2; 30]
c = fmincon(@(c)yhat(c,t)-y, c0, [1,1,1,1], 0, [0,0,1,1], 1, [],[]) 
I receive an error: 
??? Error using ==> fmincon at 647
User supplied objective function must return a scalar value.

(2) With the use of MATLAB Package bayesf
http://statmath.wu.ac.at/~fruehwirth/monographie/
Demo_mix_exponential.m  in particular
Here I fail to plug my data into the problem.

I am greatly thankful for your suggestions and help.

Answer 1

[Alan Weiss]

I am sorry, I do not understand your first constraint. You say "c(1-4)>=1" is the constraint, but this is neither mathematical nor in MATLAB notation. Perhaps you mean the sum of the values is at least 1: sum(c) >= 1. But maybe this is a complete misprint, and you really mean that all the c(i) are at least 0. You represent this as a bound constraint:

 lb = zeros(4,1);

To represent the equality constraint, you did the right thing:

 Aeq = [0,0,1,1];
 beq = 1;

You should change c0 to be feasible:

 c0 = [3,30,.5,.5];

The error you have is because your objective function does not return a scalar value. Use the following:

 fun = @(c) sum(yhat(c,t) - y).^2;

Now you are ready to proceed:

 c = fmincon(fun,c0,[],[],Aeq,beq,lb)
 plot(t,yhat(c,t),t,y)

You will see that the fit is not very good. This is primarily because your data has y(1) = .82, whereas your restriction c(3) + c(4) = 1 imply that y(1) should be 1.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[mutah]

Hi Alan Weiss plz see the MATLAB Package bayesf http://statmath.wu.ac.at/~fruehwirth/monographie/ Demo_mix_exponential.m

How i can use it ? and how I add it to my matlab ?