This is a demonstration of how to create and minimize an objective function using Pattern Search in the Genetic Algorithm and Direct Search Toolbox.
Here 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' and is taken from L.C.W. Dixon and
G.P. Szegö (eds.), Towards Global Optimisation 2, North-Holland,
Amsterdam 1978.We create an M-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 Pattern Search solver assumes the objective function will take one input x where x has as many elements as number of variables in the problem. The objective function computes the value of the function and returns that scalar value in its one return argument y.
To minimize our objective function using the PATTERNSEARCH 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] = patternsearch(ObjectiveFunction,X0)
Optimization terminated: current mesh size 9.5367e-007 is less than 'TolMesh'. x = -0.0898 0.7127 fval = -1.0316
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;
xa, 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 an M-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.
PATTERNSEARCH 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] = patternsearch(ObjectiveFunction,X0)
Optimization terminated: current mesh size 9.5367e-007 is less than 'TolMesh'. x = -0.0898 0.7127 fval = -1.0316
Consider the above function again:
f(x) = (a - b*x1^2+x1^4/3)*x1^2+x1*x2+(-c + c*x2^2)*x2^2;
By default, the PATTERNSEARCH solver only passes in one point at a time to the objective function. However, sometimes speed up can be achieved if the objective function is vectorized to take a set of points and return a set of function values.
For example if the solver wants to evaluate a set of five points in one call to this objective function, then it will call the objective with a matrix of size 5-by-2, i.e., 5 rows and 2 columns (recall 2 is the number of variables).
Create an M-file called vectorized_objective.m with the following code:
function y = vectorized_objective(x,a,b,c)
y = zeros(size(x,1),1); %Pre-allocate y
for i = 1:size(x,1) % for the number of rows in x
x1 = x(i,1);
x2 = x(i,2);
y(i) = (a - b*x1^2+x1^4/3)*x1^2+x1*x2+(-c + c*x2^2)*x2^2;
endThis vectorized version of the objective function takes a matrix x with an arbitrary number of points, the rows of x, and returns a column vector y of length the same as the number of rows of x.
To take advantage of the vectorized objective function, we need to tell PATTERNSEARCH that the objective is vectorized using the options structure that is created using PSOPTIMSET, and is passed in as the ninth argument.
ObjectiveFunction = @(x) vectorized_objective(x,4,2.1,4); X0 = [0.5 0.5]; options = psoptimset('Vectorized','on'); [x,fval] = patternsearch(ObjectiveFunction,X0,[],[],[],[],[],[],options)
Optimization terminated: current mesh size 9.5367e-007 is less than 'TolMesh'. x = -0.0898 0.7127 fval = -1.0316