## Minimize Function with Many Local Minima

### De Jong's Fifth Function

This example shows how to find a local minimum of a function using simulated annealing. The example presents two approaches for minimizing: working at the command line and using the **Optimize** Live Editor task.

De Jong's fifth function is a two-dimensional function with many (25) local minima. In the following plot, it is unclear which of these local minima is the global minimum.

dejong5fcn

Many standard optimization algorithms become stuck in local minima. Because the simulated annealing algorithm performs a wide random search, the chance of being trapped in a local minimum is decreased.

**Note**:Because simulated annealing uses random number generators, each time you run this algorithm you can get different results. See Reproduce Your Results for more information.

### Minimize at the Command Line

To run the simulated annealing algorithm without constraints, call
`simulannealbnd`

at the command line using the objective function in
`dejong5fcn.m`

, referenced by the anonymous function
`@dejong5fcn`

in the following code.

rng(10,'twister') % for reproducibility fun = @dejong5fcn; [x,fval] = simulannealbnd(fun,[0 0])

Optimization terminated: change in best function value less than options.FunctionTolerance. x = -16.1292 -15.8214 fval = 6.9034

In the results:

`x`

is the final point returned by the algorithm.`fval`

is the objective function value at the final point.

### Minimize Using the **Optimize** Live Editor Task

You can also run the minimization using the **Optimize**
Live Editor task, which provides a visual approach.

Create a new live script by clicking the

**New Live Script**button in the**File**section on the**Home**tab.Insert an

**Optimize**Live Editor task. Click the**Insert**tab and then, in the**Code**section, select**Task > Optimize**.For use in entering problem data, insert a new section by clicking the

**Section Break**button on the**Insert**tab. New sections appear above and below the task.In the new section above the task, enter the following code to define the initial point and the objective function.

x0 = [0 0]; fun = @dejong5fcn;

To place these variables into the workspace, run the section by pressing

**Ctrl+Enter**.In the

**Specify problem type**section of the task, click the**Objective > Nonlinear**button.Select

**Solver > simulannealbnd - Simulated annealing algorithm**.In the

**Select problem data**section of the task, select**Objective function > Function handle**and then choose`fun`

.Select

**Initial point (x0) > x0**.In the

**Display progress**section of the task, select the**Best value**plot.To run the solver, click the options button

**⁝**at the top right of the task window, and select**Run Section**. The plot appears in a separate figure window and in the task output area. Note that your plot might be different from the one shown, because`simulannealbnd`

is a stochastic algorithm.To see the solution and best objective function value, look at the top of the task.

The

**Optimize**Live Editor task returns variables named`solution`

and`objectiveValue`

to the workspace.To view the values these variables, enter the following code in the section below the task.

disp(solution) disp(objectiveValue)

Run the section by pressing

**Ctrl+Enter**.disp(solution)

-32.0285 -0.1280

disp(objectiveValue)

10.7632

Both the **Optimize** Live Editor
task and the command line allow you to formulate and solve problems, and they give identical
results. The command line is more streamlined, but provides less help for choosing a solver,
setting up the problem, and choosing options such as plot functions. You can also start a
problem using **Optimize**, and then generate code for command line
use, as in Solve a Constrained Nonlinear Problem, Solver-Based.