Non linear optimisation using solver based approach

I have been assigned to minimize x(1)^2+x(2)^2+x(3)^2 subject to 1-x(2)^-1*x(3)>=0,x(1)-x(3)>=0,x(1)-x(2)^2+x(2)*x(3)-4=0,0<=x(1)<=5,0<=x(2)<=3,0<=x(3)<=3.

I have written the following script for the above optimization problem:

% Solver based approach for non linear problem

%define the objective function

f=@(x) (x(1)^2+x(2)^2+x(3)^2);

%initial feasible solution

x0=[0 0 0];

A=[0 -1 1;-1 0 1];b=[0;0];%linear inequality constraints

Aeq=[];beq=[];

lb=[0;0;0];ub=[5;3;3];%define the bounds

c=[];%there is no non linear inequality

ceq=@(x) (x(1)-x(2)^2+x(2)*x(3)-4);%non linear equality constraint

nlcon=@(x) deal(c,ceq(x));

[x,fval]=fmincon(f,x0,A,b,Aeq,beq,lb,ub,nlcon)

I am getting the values 4,0.0003,0.0002 and these values meet all the conditions except one which is x(1)-x(2)^2+x(2)*x(3)-4=0. With the above values, this expression becomes -0.00000003 which is not 0 obviously

Kindly check and give feedback and suggest corrections

Answer 1

[Gayatri]

Hi Bibhudatta,

• Given the optimization problem and the MATLAB script, it appears that you have encountered a common issue in numerical optimization which is the tolerance of the solution. This discrepancy is likely due to the tolerance settings of the ‘fmincon’ solver.
• You can adjust the tolerance settings of ‘fmincon’ using the ‘optimoptions' function.

Here's how you can adjust the tolerance settings in your script:

% Define the objective function
f = @(x) (x(1)^2 + x(2)^2 + x(3)^2);
% Initial feasible solution
x0 = [0, 0, 0];
% Linear inequality constraints
A = [0, -1, 1; -1, 0, 1];
b = [0; 0];
% Bounds
lb = [0; 0; 0];
ub = [5; 3; 3];

% Nonlinear equality constraint
ceq = @(x) (x(1) - x(2)^2 + x(2)*x(3) - 4);
nlcon = @(x) deal([], ceq(x));
% Create options with a tighter constraint tolerance
options = optimoptions('fmincon', 'ConstraintTolerance', 1e-10);
% Solve the optimization problem with the specified options
[x, fval] = fmincon(f, x0, A, b, [], [], lb, ub, nlcon, options)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1x3

4.0000 0.0001 0.0001

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

fval = 16.0000

Please refer the below documentation for ‘optimoptions’ function:

https://in.mathworks.com/help/optim/ug/optim.problemdef.optimizationproblem.optimoptions.html

I Hope it helps!

[Bibhudatta]

Thanks Gayatri , this really helped .