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This example shows how to solve a nonlinear minimization problem with tridiagonal Hessian matrix approximated by sparse finite differences instead of explicit computation.

The problem is to find *x* to minimize

$$f(x)={\displaystyle \sum _{i=1}^{n-1}\left({\left({x}_{i}^{2}\right)}^{\left({x}_{i+1}^{2}+1\right)}+{\left({x}_{i+1}^{2}\right)}^{\left({x}_{i}^{2}+1\right)}\right)},$$

where *n* = `1000`

.

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 `SpecifyObjectiveGradient`

option
to `true`

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,'Algorithm','trust-region',... 'SpecifyObjectiveGradient',true,'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 = 7.47389-17 output = struct with fields: iterations: 7 funcCount: 8 stepsize: 0.0046 cgiterations: 7 firstorderopt: 7.9822e-10 algorithm: 'trust-region' message: 'Local minimum found.…' constrviolation: []

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