Minimize problem using PSO

I want to find a minimum value of y(and the corresponding value of L and x), for various values of L and x. How to do this using Particle Swarm Optimization ?

The equation is:

Y= (20+500*L)/x^(1.2+(3*L))

Answer 1

[Sam Chak]

Hi @Reji G

It seems that the function does not have any global minima.

[X, L] = meshgrid(1:3/40:4, 0.1:0.3/40:0.4);
Y = (20 + 500*L)./X.^(1.2 + 3*L);
surf(X, L, Y)
xlabel('x'); ylabel('L'); zlabel('y'); 
view(45, 30)

% Using PSO to minimize the function with the specified bound constraints
f     = @(x) (20 + 500*x(2))./x(1).^(1.2 + 3*x(2));
nvars = 2;
lb    = [1 0.1];    % lower bounds
ub    = [4 0.4];    % upper bounds
[x, fval] = particleswarm(f, nvars, lb, ub)

Optimization ended: relative change in the objective value 
over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

x = 1×2

4.0000 0.4000

fval = 7.8973

[Reji G]

Hello,

Thank you for helping.

I want MATLAB to display the minimum value of y and the corresponding value of x and L in which minimum occur. I don't need any graphical representaion. See the image. Ex:- If C8 is minimum, then MATLAB should display the minimum value of y and corresponding x and L values. This should be implemented stictly using "PARTICLE SWARM OPTIMIZATION". Thanks in advance.

[Sam Chak]

Hi @Reji G,

Don't mention it. It's just a simple help. Please ignore the graphical representation. The Answer is updated with the PSO demonstration of minimizing the given function within the bounds {}, {} as shown in your image. The minimum point is strictly found and displayed by PSO as requested, regardless of the clues from the graphical representation.

[Reji G]

Hello Sam,

Its working. Thank you. Need a small clarification.

1. How to know its displaying minimum value?
2. What changes need to be done in this program to display maximum value?
3. How to list all the values?(for all x and L combinations. c1 to c15 in case of my image)

[Sam Chak]

Hi Reji,

1. The final objective function value of swarm particles is displayed as fval which indicates the function value at best solution found so far.

2. This is how to perform the maximization:

% Using PSO to maximize the function with the specified bound constraints
f     = @(x) (20 + 500*x(2))./x(1).^(1.2 + 3*x(2));
fmax  = @(x) -f(x);
nvars = 2;
lb    = [1 0.1];    % lower bounds
ub    = [4 0.4];    % upper bounds
[x, fval] = particleswarm(fmax, nvars, lb, ub)

Optimization ended: relative change in the objective value 
over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

x = 1×2

1.0000 0.4000

fval = -220

fmaxValue = -fval   % can verify the result with the graphical representation

fmaxValue = 220

3. Although the algorithm performs the search within the bound constraints, it does NOT search the entire space as in to (as in your scenario). It is possible list/store the position and the objective function value of each swarm particle in each iteration through calling the OutputFcn, something like this:

options   = optimoptions(@particleswarm, 'OutputFcn', @pswoutfun)
[x, fval] = particleswarm(fmax, nvars, lb, ub, options)

where you have write the code for the pswoutfun.m file. But it can be a little tedious to write the code here. You can find some templates in

edit pswplotbestf
edit psoutputfile

For more info, please check:

https://www.mathworks.com/help/gads/particle-swarm-options.html#bued1tk-1

If you find this tutorial on using particleswarm() is helpful, consider accepting ✔ and voting 👍 the Answer. Thanks, @Reji G!