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To use Global Optimization Toolbox functions,
first write a file (or an anonymous function) that computes the function
you want to optimize. This is called an objective function for most
solvers, or fitness function for `ga`

. The function
should accept a vector, whose length is the number of independent
variables, and return a scalar. For `gamultiobj`

,
the function should return a row vector of objective function values.
For vectorized solvers, the function should accept a matrix, where
each row represents one input vector, and return a vector of objective
function values. This section shows how to write the file.

This example shows how to write a file for the function you want to optimize. Suppose that you want to minimize the function

$$f(x)={x}_{1}^{2}-2{x}_{1}{x}_{2}+6{x}_{1}+4{x}_{2}^{2}-3{x}_{2}.$$

The file that computes this function must accept a vector `x`

of
length 2, corresponding to the variables *x*_{1} and *x*_{2},
and return a scalar equal to the value of the function at `x`

.

Select

**New > Script**(**Ctrl+N**) from the MATLAB^{®}**File**menu. A new file opens in the editor.Enter the following two lines of code:

function z = my_fun(x) z = x(1)^2 - 2*x(1)*x(2) + 6*x(1) + 4*x(2)^2 - 3*x(2);

Save the file in a folder on the MATLAB path.

Check that the file returns the correct value.

my_fun([2 3]) ans = 31

For `gamultiobj`

, suppose you have three
objectives. Your objective function returns a three-element vector
consisting of the three objective function values:

function z = my_fun(x) z = zeros(1,3); % allocate output z(1) = x(1)^2 - 2*x(1)*x(2) + 6*x(1) + 4*x(2)^2 - 3*x(2); z(2) = x(1)*x(2) + cos(3*x(2)/(2+x(1))); z(3) = tanh(x(1) + x(2));

The `ga`

, `gamultiobj`

,
and `patternsearch`

solvers optionally
compute the objective functions of a collection of vectors in one
function call. This method can take less time than computing the objective
functions of the vectors serially. This method is called a vectorized
function call.

To compute in vectorized fashion:

Write your objective function to:

Accept a matrix with an arbitrary number of rows.

Return the vector of function values of each row.

For

`gamultiobj`

, return a matrix, where each row contains the objective function values of the corresponding input matrix row.

If you have a nonlinear constraint, be sure to write the constraint in a vectorized fashion. For details, see Vectorized Constraints.

Set the

`UseVectorized`

option to`true`

using`optimoptions`

, or set**User function evaluation > Evaluate objective/fitness and constraint functions**to`vectorized`

in the Optimization app. For`patternsearch`

, also set`UseCompletePoll`

to`true`

. Be sure to pass the options structure to the solver.

For example, to write the objective function of Write a Function File in a vectorized fashion,

function z = my_fun(x) z = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ... 4*x(:,2).^2 - 3*x(:,2);

To use `my_fun`

as a vectorized objective function
for `patternsearch`

:

options = optimoptions('patternsearch','UseCompletePoll',true,'UseVectorized',true); [x fval] = patternsearch(@my_fun,[1 1],[],[],[],[],[],[],... [],options);

To use `my_fun`

as a vectorized objective function
for `ga`

:

options = optimoptions('ga','UseVectorized',true); [x fval] = ga(@my_fun,2,[],[],[],[],[],[],[],options);

For `gamultiobj`

,

function z = my_fun(x) z = zeros(size(x,1),3); % allocate output z(:,1) = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ... 4*x(:,2).^2 - 3*x(:,2); z(:,2) = x(:,1).*x(:,2) + cos(3*x(:,2)./(2+x(:,1))); z(:,3) = tanh(x(:,1) + x(:,2));

To use `my_fun`

as a vectorized objective function
for `gamultiobj`

:

options = optimoptions('ga','UseVectorized',true); [x fval] = gamultiobj(@my_fun,2,[],[],[],[],[],[],options);

For more information on writing vectorized functions for `patternsearch`

,
see Vectorize the Objective and Constraint Functions. For more
information on writing vectorized functions for `ga`

,
see Vectorize the Fitness Function.

If you use `GlobalSearch`

or `MultiStart`

,
your objective function can return derivatives (gradient, Jacobian,
or Hessian). For details on how to include this syntax in your objective
function, see Including Gradients and Hessians in the Optimization Toolbox™ documentation.
Use `optimoptions`

to set options
so that your solver uses the derivative information:

**Local Solver = fmincon, fminunc**

Condition | Option Setting |
---|---|

Objective function contains gradient | `'SpecifyObjectiveGradient' = true` ; see How to Include Gradients |

Objective function contains Hessian | `'HessianFcn' = 'objective'` or a function
handle; see Including Hessians |

Constraint function contains gradient | `'SpecifyConstraintGradient' = true` ; see Including Gradients in Constraint Functions |

**Local Solver = lsqcurvefit, lsqnonlin**

Condition | Option Setting |
---|---|

Objective function contains Jacobian | `'SpecifyObjectiveGradient' = true` |

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