fmincon, the size of the current step is less than the value of the step size tolerance, but constraints are not satisfied

hi, I am working on a non-linear optimization problem.

I try to find the upper bound with fmincon, then currently facing this problem:

fmincon stopped because the size of the current step is less than the value of the step size tolerance but constraints are not satisfied to within the value of the constraint tolerance.

And I have to formulated it in the following way

Here is my current code:

H = zeros(4);
H(3,3) = 1; H(4,4) = 1;
c = [1 1 0 0]';
Q1 = zeros(4); Q1(1, 2) = 0.5; Q1(2, 1) = 0.5; Q1(2, 3) = 0.5; Q1(3, 2) = 0.5;
Q2 = zeros(4); Q2(1, 2) = 0.5; Q2(2, 1) = 0.5;
Q3 = zeros(4); Q3(2, 3) = 0.5; Q3(3, 2) = 0.5;
Q = [Q1; Q2; Q3];
A = zeros(3,4); A(2,4) = 1; A(3,1) = 1;
b = [2 3 5]';
lb = [0 0 0 0]'; ub = [10 4 10 10]';

objective = @(x) x'*H*x + c'*x;
x_0 = 0.5 * (lb + ub); % Use average as initial value
A_nlp = [];   b_nlp = [];
Aeq_nlp = []; beq_nlp = [];
function [c, c_eq] = NonLinearConstraint(x)
    c = []; % No nonlinear inequalities    
    for i = 1:m % Loop over the matrices defining the quadratic constraints
        Qi = Q((i-1)*n+1:i*n,:);
        c_eq(i)= x'*Qi*x - A(i,:)*x - b(i);
    end
end
nonlcon = @NonLinearConstraint;
options = optimoptions('fmincon','Display','iter','Algorithm','sqp');

[~, fval] = fmincon(objective, x_0, A_nlp, b_nlp, Aeq_nlp, beq_nlp, lb, ub, nonlcon, options);

It could not satisfy the constraint function. I also tried a few other optimoptions (different algorithm, step size ..)

Could anyone give me some advice how to fix this?

Thank you for your time!

[Walter Roberson]

The minimum is 4 at [2, 1, 0, 1]

Your objective should be x1 + x2 + x3^2 + x4^2 and you should put in linear and nonlinear constraints and upper and lower bounds.

[Huu Duc Nguyen]

Hi, thank you for your reply.

As I check, [2 1 0 1] actually doesn't satisfy the 3rd equality constraint

x1 + x2 * x3 = 5

With the simple code to this, I got the right solution.

objective = @(x) x(1) + x(2) + x(3)^2 + x(4)^2;
lb = [0 0 0 0]';
ub = [10 4 10 10]';
x_0 = 0.5 * (lb + ub);
A_nlp = [];
b_nlp = [];
Aeq_nlp = [];
beq_nlp = [];

nonlcon = @NonLinearConstraint;
options = optimoptions('fmincon','Display','iter');

[~, f_ub,~,output] = fmincon(objective, x_0, A_nlp, b_nlp, Aeq_nlp, beq_nlp, lb, ub, nonlcon, options);
display(f_ub)

function [c, c_eq] = NonLinearConstraint(x)
    c = [];
    c_eq(1) = x(1)*x(2) + x(2)*x(3) - 2;
    c_eq(2) = x(1)*x(2) + x(4) - 3;
    c_eq(3) = x(1) + x(2)*x(3) - 5;    
end

However, I am required to formulate in a generalized approach, which leads to the upper error.

[Bruno Luong]

"The minimum is 4 at [2, 1, 0, 1] "

But

x(1) + x(2)*x(3) is 2 not 5 as required

[Walter Roberson]

Darn, all that work I did earlier, I had accidentally entered

x1 + x2*3 == 5

Answer 1

[Bruno Luong]

I beleive the correct sign of A is "+"

c_eq(i)= x'*Qi*x + A(i,:)*x - b(i);

[Huu Duc Nguyen]

Oh my god, you are absolutely right :3

Thank you for pointing out that!!