Unconstrained Minimization Using fminunc

This example shows how to use fminunc to solve the nonlinear minimization problem

$$\underset{x}{\min }f(x)=e^{x_1}\left(4x_1^2+2x_2^2+4x_1{x}_2+2x_2+1\right).$$

To solve this two-dimensional problem, write a function that returns $$f(x)$$. Then, invoke the unconstrained minimization routine fminunc starting from the initial point x0 = [-1,1].

The helper function objfun at the end of this example calculates $$f(x)$$.

To find the minimum of $$f(x)$$, set the initial point and call fminunc.

x0 = [-1,1];
[x,fval,exitflag,output] = fminunc(@objfun,x0);

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

<stopping criteria details>

View the results, including the first-order optimality measure in the output structure.

disp(x)

    0.5000   -1.0000

disp(fval)

   3.6609e-15

disp(exitflag)

     1

disp(output.firstorderopt)

   1.2284e-07

The exitflag output indicates whether the algorithm converges. exitflag = 1 means fminunc finds a local minimum.

The output structure gives more details about the optimization. For fminunc, the structure includes:

• output.iterations, the number of iterations

• output.funcCount, the number of function evaluations

• output.stepsize, the final step-size

• output.firstorderopt, a measure of first-order optimality (which, in this unconstrained case, is the infinity norm of the gradient at the solution)

• output.algorithm, the type of algorithm used

• output.message, the reason the algorithm stopped

Helper Function

This code creates the objfun helper function.

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);
end