How do I include or make a discrete variable for an optimization problrm?

This is the code I made and I got a values: x1=3 x2=8 x3=3 x4=3 x5=0 x6=0 x7=11

but x1 result is not in the group of numbers {2,5,7,8}

please help here is the code

clear all;
clc;

% Define the objective function coefficients
f = [2; -6; 3; 4; 7; 3; 2];

% Define the inequality constraints in matrix form
A = [0 -1 -5 -3 1 -2 -3;
    6 3 2 -1 -2 2 0;
     -7 -4 3 -4 3 -2 -1];

% Define the right-hand side of the inequality constraints
b = [-65; 73; -66];

% Define the lower and upper bounds for each variable
lb = [2; 2; 3; 3; 0; 0; -Inf];
ub = [8; 8; 6; 6; Inf; Inf; Inf];

% Define integer variables (1 for discrete variables, 0 for continuous)
intcon = [1 2 3 4]; % Only x1 and x2 are discrete

% Solve the mixed-integer linear programming problem
options = optimoptions('intlinprog', 'Display', 'off');
[x, fval, exitflag] = intlinprog(f, intcon, A, b, [], [], lb, ub, options);

% Display the optimal solution and objective function value
if exitflag == 1
    disp('Optimal Solution:')
    disp(x);
    disp('Optimal Objective Function Value:')
    disp(fval);
else
    disp('No optimal solution found.');
end

Answer 1

[Bruno Luong]

To restrict x1 to { 2, 5, 7, 8 }

You might make slack variables

x1 = 2 + 3*y1 + 2*y2 + 1*y3;

with constrants

y1, y2, y3 integers in {0, 1} and 0 <= y3 <= y2 <= y1

So the same thing for x2, x3, x4, then optimize wrt (x5, x6, x7, y1, y2 ...)

Answer 2

[Torsten]

Make a loop over the 4*4*3*3 = 144 possible combinations of x1,x2,x3 and x4 and only optimize with respect to x5, x6 and x7.

In this case, you can use "linprog" to solve the subproblems.

Of course - if problems become larger - this way of solving will have its limits.