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This example shows how to create and minimize an objective function using Simulated Annealing in the Global Optimization Toolbox.

We want to minimize a simple function of two variables

min f(x) = (4 - 2.1*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-4 + 4*x2^2)*x2^2; x

The above function is known as 'cam' as described in L.C.W. Dixon and G.P. Szego (eds.), Towards Global Optimisation 2, North-Holland, Amsterdam, 1978.

We create a MATLAB file named simple_objective.m with the following code in it:

function y = simple_objective(x) y = (4 - 2.1*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ... (-4 + 4*x(2)^2)*x(2)^2;

The Simulated Annealing solver assumes the objective function will take one input x where x has as many elements as the number of variables in the problem. The objective function computes the scalar value of the objective and returns it in its single return argument y.

To minimize our objective function using the SIMULANNEALBND function, we need to pass in a function handle to the objective function as well as specifying a start point as the second argument.

```
ObjectiveFunction = @simple_objective;
X0 = [0.5 0.5]; % Starting point
[x,fval,exitFlag,output] = simulannealbnd(ObjectiveFunction,X0)
```

Optimization terminated: change in best function value less than options.FunctionTolerance. x = -0.0896 0.7130 fval = -1.0316 exitFlag = 1 output = struct with fields: iterations: 2948 funccount: 2971 message: 'Optimization terminated: change in best function value less than options.FunctionTolerance.' rngstate: [1x1 struct] problemtype: 'unconstrained' temperature: [2x1 double] totaltime: 2.3698

The first two output arguments returned by SIMULANNEALBND are `x`

, the best point found, and `fval`

, the function value at the best point. A third output argument, `exitFlag`

returns a flag corresponding to the reason SIMULANNEALBND stopped. SIMULANNEALBND can also return a fourth argument, `output`

, which contains information about the performance of the solver.

SIMULANNEALBND can be used to solve problems with bound constraints. The lower and upper bounds are passed to the solver as vectors. For each dimension i, the solver ensures that lb(i) <= x(i) <= ub(i), where x is a point selected by the solver during simulation. We impose the bounds on our problem by specifying a range -64 <= x(i) <= 64 for x(i).

lb = [-64 -64]; ub = [64 64];

Now, we can rerun the solver with lower and upper bounds as input arguments.

[x,fval,exitFlag,output] = simulannealbnd(ObjectiveFunction,X0,lb,ub); fprintf('The number of iterations was : %d\n', output.iterations); fprintf('The number of function evaluations was : %d\n', output.funccount); fprintf('The best function value found was : %g\n', fval);

Optimization terminated: change in best function value less than options.FunctionTolerance. The number of iterations was : 2428 The number of function evaluations was : 2447 The best function value found was : -1.03163

Simulated annealing mimics the annealing process to solve an optimization problem. It uses a temperature parameter that controls the search. The temperature parameter typically starts off high and is slowly "cooled" or lowered in every iteration. At each iteration a new point is generated and its distance from the current point is proportional to the temperature. If the new point has a better function value it replaces the current point and iteration counter is incremented. It is possible to accept and move forward with a worse point. The probability of doing so is directly dependent on the temperature. This unintuitive step sometime helps identify a new search region in hope of finding a better minimum.

Sometimes we want our objective function to be parameterized by extra arguments that act as constants during the optimization. For example, in the previous objective function, say we want to replace the constants 4, 2.1, and 4 with parameters that we can change to create a family of objective functions. We can re-write the above function to take three additional parameters to give the new minimization problem.

min f(x) = (a - b*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-c + c*x2^2)*x2^2; x

a, b, and c are parameters to the objective function that act as constants during the optimization (they are not varied as part of the minimization). One can create a MATLAB file called parameterized_objective.m containing the following code.

function y = parameterized_objective(x,a,b,c) y = (a - b*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ... (-c + c*x(2)^2)*x(2)^2;

Again, we need to pass in a function handle to the objective function as well as a start point as the second argument.

SIMULANNEALBND will call our objective function with just one argument `x`

, but our objective function has four arguments: x, a, b, c. We can use an anonymous function to capture the values of the additional arguments, the constants a, b, and c. We create a function handle 'ObjectiveFunction' to an anonymous function that takes one input `x`

, but calls 'parameterized_objective' with x, a, b and c. The variables a, b, and c have values when the function handle 'ObjectiveFunction' is created, so these values are captured by the anonymous function.

```
a = 4; b = 2.1; c = 4; % define constant values
ObjectiveFunction = @(x) parameterized_objective(x,a,b,c);
X0 = [0.5 0.5];
[x,fval] = simulannealbnd(ObjectiveFunction,X0)
```

Optimization terminated: change in best function value less than options.FunctionTolerance. x = 0.0898 -0.7127 fval = -1.0316

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