Optimization Toolbox

Unconstrained Example

Consider the problem of finding a set of values [x₁, x₂] that solves

(1-1)

To solve this two-dimensional problem, write an M-file that returns the function value. Then, invoke the unconstrained minimization routine fminunc.

Step 1: Write an M-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);

Step 2: Invoke one of the unconstrained optimization routines

x0 = [-1,1];    % Starting guess
options = optimset('LargeScale','off');
[x,fval,exitflag,output] = fminunc(@objfun,x0,options);

After 40 function evaluations, this produces the solution

x =
      0.5000   -1.0000

The function at the solution x is returned in fval.

fval =
      1.3030e-10

The exitflag tells if the algorithm converged. An exitflag > 0 means a local minimum was found.

exitflag =
     1

The output structure gives more details about the optimization. For fminunc, it includes the number of iterations in iterations, the number of function evaluations in funcCount, the final step-size in stepsize, a measure of first-order optimality (which in this unconstrained case is the infinity norm of the gradient at the solution) in firstorderopt, and the type of algorithm used in algorithm.

output = 
       iterations: 7
        funcCount: 40
         stepsize: 1
    firstorderopt: 9.2801e-004
        algorithm: 'medium-scale: Quasi-Newton line search'

When more than one local minimum exists, the initial guess for the vector [x₁, x₂] affects both the number of function evaluations and the value of the solution point. In the example above, x0 is initialized to [-1,1].

The variable options can be passed to fminunc to change characteristics of the optimization algorithm, as in

x = fminunc(@objfun,x0,options);

options is a structure that contains values for termination tolerances and algorithm choices. An options structure can be created using the optimset function

options = optimset('LargeScale','off');

In this example we have turned off the default selection of the large-scale algorithm and so the medium-scale algorithm is used. Other options include controlling the amount of command line display during the optimization iteration, the tolerances for the termination criteria, if a user-supplied gradient or Jacobian is to be used, and the maximum number of iterations or function evaluations. See optimset, the individual optimization functions, and Table 4-3, Optimization Options Parameters, for more options and information.

Examples that Use Standard Algorithms

Nonlinear Inequality Constrained Example