## Documentation Center |

Next, solve the same problem but the Hessian matrix is now approximated
by sparse finite differences instead of explicit computation. To use
the `trust-region` method in `fminunc`,
you *must* compute the gradient in `fun`;
it is *not* optional as in the `quasi-newton` method.

The `brownfg` file computes the objective function
and gradient.

This function file ships with your software.

function [f,g] = brownfg(x) % BROWNFG Nonlinear minimization test problem % % Evaluate the function n=length(x); y=zeros(n,1); i=1:(n-1); y(i)=(x(i).^2).^(x(i+1).^2+1) + ... (x(i+1).^2).^(x(i).^2+1); f=sum(y); % Evaluate the gradient if nargout > 1 if nargout > 1 i=1:(n-1); g = zeros(n,1); g(i) = 2*(x(i+1).^2+1).*x(i).* ... ((x(i).^2).^(x(i+1).^2))+ ... 2*x(i).*((x(i+1).^2).^(x(i).^2+1)).* ... log(x(i+1).^2); g(i+1) = g(i+1) + ... 2*x(i+1).*((x(i).^2).^(x(i+1).^2+1)).* ... log(x(i).^2) + ... 2*(x(i).^2+1).*x(i+1).* ... ((x(i+1).^2).^(x(i).^2)); end

To allow efficient computation of the sparse finite-difference
approximation of the Hessian matrix *H*(*x*),
the sparsity structure of *H* must be predetermined.
In this case assume this structure, `Hstr`, a sparse
matrix, is available in file `brownhstr.mat`. Using
the `spy` command you can see
that `Hstr` is indeed sparse (only 2998 nonzeros).
Use `optimoptions` to set the `HessPattern` option
to `Hstr`. When a problem as large as this has obvious
sparsity structure, not setting the `HessPattern` option
requires a huge amount of unnecessary memory and computation because `fminunc` attempts to use finite differencing
on a full Hessian matrix of one million nonzero entries.

You must also set the `GradObj` option to `'on'` using `optimoptions`, since the gradient is computed
in `brownfg.m`. Then execute `fminunc` as
shown in Step 2.

fun = @brownfg; load brownhstr % Get Hstr, structure of the Hessian spy(Hstr) % View the sparsity structure of Hstr

n = 1000; xstart = -ones(n,1); xstart(2:2:n,1) = 1; options = optimoptions(@fminunc,'GradObj','on','HessPattern',Hstr); [x,fval,exitflag,output] = fminunc(fun,xstart,options);

This 1000-variable problem is solved in seven iterations and
seven conjugate gradient iterations with a positive `exitflag` indicating
convergence. The final function value and measure of optimality at
the solution `x` are both close to zero (for `fminunc`, the first-order optimality is
the infinity norm of the gradient of the function, which is zero at
a local minimum):

exitflag,fval,output exitflag = 1 fval = 7.4738e-17 output = iterations: 7 funcCount: 8 cgiterations: 7 firstorderopt: 7.9822e-10 algorithm: 'large-scale: trust-region Newton' message: 'Local minimum found. Optimization completed because the size of the grad...' constrviolation: []

Was this topic helpful?