how to use the optimization toolbox to optimize two varibles multiple times in matlab

To optimize two variables multiple times in Matlab using the Optimization Toolbox, you can make use of the multiobjective function along with the optimoptions function.

Here's an example code snippet that shows how to optimize the two variables x and y for a given objective function fun 10 times:

main.m
% Define Objective Function
fun = @(x) [x(1)^2 + x(2)^2, (x(1)-1)^2 + x(2)^2];

% Define Lower Bound and Upper Bound of Variables
lb = [-10, -10];
ub = [10, 10];

% Set Optimization Options
options = optimoptions('multiobjective', 'Display', 'iter', 'MaxIterations', 10);

% Optimize the Variables
[x, fval] = multiobjective(fun, 2, [], [], [], [], lb, ub, options);
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In this example, fun is the objective function that takes a 2-element vector x as input and returns a 2-element vector as output. lb and ub denote the lower and upper bounds of x respectively. The MaxIterations option specifies the maximum number of iterations for the optimization process. Finally, the multiobjective function is called with the objective function fun, number of objective functions as 2, empty matrices for the linear and nonlinear constraints, bounds of the variables lb and ub, and the optimization options options. The function returns the optimized variables x and their corresponding function values fval.