Page 239 - Numerical Methods for Chemical Engineering

228 5 Numerical optimization

Unconstrained minimizer fminunc in MATLAB

The routine fminunc, part of the optional MATLAB optimization toolkit, uses either a
gradient or a Newton algorithm to solve an unconstrained, continuous optimization problem,
with the choice of method depending upon the size of the problem and upon the level of
information about the gradient and Hessian provided by the user. It is called with the syntax
[x, F, exitflag, output] = fminunc (fun, x0, OPTIONS, P1, P2,...);

fun is the name of a routine that returns the cost function value,
function F = fun(x, P1, P2,...);
x0 is the initial guess, and P1, P2, . . . are optional parameters to be passed through fminunc
to fun. OPTIONS is set by optimset.If fun supplies the value of the gradient as well, use
OPTIONS = optimset(‘GradObj’, ‘on’);

and give fun the syntax,
function [F, grad] = fun(x, P1, P2,...);
If in addition, you supply the Hessian matrix, add to OPTIONS,
OPTIONS = optimset(OPTIONS, ‘Hessian’, ‘on’);
and write fun as

function [F, grad, Hessian] = fun(x, P1, P2,...);
The output arguments include the local minimum x and its cost function value F.
For large problems, fminunc can be informed of the sparsity pattern of the Hessian
through the ‘HessPattern’ field. Instead of supplying the Hessian itself, one can merely pass
the name of a routine that computes the product of an input vector with the Hessian through
the ‘HessMult’ field. fminunc can be told explicitly to use algorithms effective for large
problems by setting the ‘LargeScale’ field to ‘on’. The ‘Display’ field controls the level of
detail printed to the screen about the progress of the minimization and ‘Diagnostics’ can be
set to ‘on’ to gain further information about the optimizer performance.

Example. A simple cost function
We demonstrate fminunc for the simple cost function (5.15) used to compare the per-
formance of the steepest descent and conjugate gradient methods. The routine sim-
ple cost func.m that returns the value of this cost function was presented previously fol-
lowing (5.15). The code below uses fminunc to find the minimum at (1, 2),

x0 = [5;0];
ModelParam.c = 1;
[x,F] = fminunc (@simple cost func, x0, [], ModelParam),
Warning: Gradient must be provided for trust-region method;
using line-search method instead.
> In fminunc at 241