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This example shows how to solve a nonlinear minimization problem with an explicit tridiagonal Hessian matrix H(x).
The problem is to find x to minimize
$$f(x)={\displaystyle \sum _{i=1}^{n-1}\left({\left({x}_{i}^{2}\right)}^{\left({x}_{i+1}^{2}+1\right)}+{\left({x}_{i+1}^{2}\right)}^{\left({x}_{i}^{2}+1\right)}\right)},$$ | (6-16) |
where n = 1000.
The file is lengthy so is not included here. View the code with the command
type brownfgh
Because brownfgh computes the gradient and Hessian values as well as the objective function, you need to use optimoptions to indicate that this information is available in brownfgh, using the GradObj and Hessian options.
n = 1000; xstart = -ones(n,1); xstart(2:2:n,1) = 1; options = optimoptions(@fminunc,'Algorithm','trust-region',... 'GradObj','on','Hessian','on'); [x,fval,exitflag,output] = fminunc(@brownfgh,xstart,options);
This 1000 variable problem is solved in about 7 iterations and 7 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:
fval,exitflag,output fval = 2.8709e-17 exitflag = 1 output = iterations: 7 funcCount: 8 cgiterations: 7 firstorderopt: 4.7948e-10 algorithm: 'trust-region' message: 'Local minimum found. Optimization completed because the size o...' constrviolation: []