Matlab: need help with optimization

Minimize the following objective function over $(x_1,x_2)$

$$f (x_1, x_2) := \exp(x_1) \, (4 x_1^2 + 2 x_2^2 + 4 x_1 x_2 + 2 x_2 + 1) + b$$

subject to the constraints

$$x_1^2 + x_2 = 1 - b \qquad \qquad -x_1 x_2 \leq 10$$

Note that $b$ is also in one of the constraints.

I want to run the optimization 20 times, for b=1, b=2, b=3.....b=20 ( because b is between 0 to 20). I also want to store the optimal (x(1),x(2),b) so that I can plot it later.

There are existing reference ((http://www.mathworks.com/help/optim/ug/nonlinear-equality-and-inequality-constraints.html)) which I followed. How do I modify the following code so that it will do the above?

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+b;


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);


Update:

I tried to modified my objective function by adding the b,

function f = objfun(x,b)
f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b;


and I got

    Error using objfun (line 2)
Not enough input arguments.Error in fmincon (line 640)

initVals.f = feval(funfcn{3},X,varargin{:});
Caused by:
Failure in initial user-supplied objective function evaluation. FMINCON cannot     continue.

• You could notice that, from the equality constraint, $x_2$ can be expressed as a function of $x_1$ and $b$ and that the inequality constraint defines the range for $x_1$. I suppose that this could make your problem easier. Aug 25, 2014 at 6:38

I haven't tested this, but something like it should work.

First, modify objfun so that b is an input arg. Then try this:

cnt = 1;
xopt = [];
fval = [];
for b = 1 : 20

cfun = @(x)objfun(x,b);
[xopt(cnt),fval(cnt)] = fmincon(cfun,[-1 1],[],[],[],[],[],[],...
@confuneq,options);

cnt = cnt + 1;

end

• how should I modify "objfun" m-file to include the b input?
– kuku
Oct 26, 2014 at 2:42
• Sorry. My answer should have passed in cfun to fmincon and not @objfun. I fixed it. Let me know if this works or not. Oct 27, 2014 at 1:21