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I am trying to solve a multiobjective optimization problem.

I have 4 objective functions each of which is a function in 3 variables. Among the 3 variables, 2 variables need to be optimized and are to be used to calculate the 3rd variable using a closed form equation. However, I am unable to figure out how to optimize only 2 of the 3 variables.

Description of overall problem

Each objective function is derived from different inputs which are known. Lets assume inputs as "input_1", "input_2", "input_3", "input_4"

Let's assume the variables to be optimized are "x", "y" and the variables to be calculated are "theta_i", where i=1,2,3,4.

objective function 1 = function (input_1, x,y, theta_1)

objective function 2 = function (input_2, x,y, theta_2)

objective function 3 = function (input_3, x,y, theta_3)

objective function 4 = function (input_4, x,y, theta_4)

optimization solution = gamultiobj([ob1, ob2, ob3, ob4],.....,options);

also

theta_1 = different_function(input_1,x,y)

theta_2 = different_function(input_2,x,y)

theta_3 = different_function(input_3,x,y)

theta_4 = different_function(input_4,x,y)

different_function is not known and needs to be calculated but is generally of the form as stated above.

Any advice on how to proceed with this problem is highly appreciated.

Fabio Freschi
on 9 Jun 2020

Why don't you remove theta from your objective function definition and call different_function inside function?

function objective_function_1 = objfunction1(input_1, x,y)

theta_1 = different_function(input_1,x,y);

% interesting stuff with input_1, x,y, theta_1

...

end

Fabio Freschi
on 9 Jun 2020

In this case you can set the upper and lower bound for that specific variable to the same value. Maybe it is not efficient but it works. Look the code below

% objective functions

fun = @(x)[x(:,1).^2+x(:,2).^2; (x(:,1)-1).^2+x(:,2).^2];

% number of vars

nvars = 2;

% upper and lower bounds

lb = [-2; -2], ub = [2; 2];

% run optimization

x = gamultiobj(fun,nvars,[],[],[],[],lb,ub);

% set x(:,2) to 0 with same lower and upper bounds

lb = [-2; 0], ub = [2; 0];

% run optimization

x = gamultiobj(fun,nvars,[],[],[],[],lb,ub)

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