Does the order of magnitude of the variables in the non linear constraints influence the solution in fmincon?

Can you please help me to figure out the problem with this optimization ? Starting from the given initial point, I obtain the message :

'Solver stopped prematurely.↵↵fmincon stopped because it exceeded the function evaluation limit,↵options.MaxFunctionEvaluations = 3000 (the default value).'

When I use the given value for f, the constraint is no longer defined in the considered point and I get the following message

'Error using barrier Nonlinear constraint function is undefined at initial point. Fmincon cannot continue.'

An interesting remark is that that value of the initial point x0=0.6*b is a local minimum for the objective function @(f)0 and with the same non linear constraints when C_Ter=5*1e+11.

Does this refer to a problem of order of magnitude of my variables?

clear all;
%%%%%%% simulation parameters

n=20 
m=5; 
power_BS = 20 
N0=10e-10;
fb = 0.18*1e+6  
n_RB = 100 
bandwidth= 100*1e+6;  
b_v=fb*n_RB/n;
radius_BS = 500; 
b=b_v*rand(n,1); 
d_sq=radius_BS*rand(n+m,1).^2;
h=exprnd(1,n+m,1)./(d_sq);  
SNR=h*power_BS/N0;
 
C_Ter=5*1e+6;
p=[];

x0=[52.8827;   45.0967;   45.3726;   46.4245;   53.6886;  1.2579*ones(15,1)] % initial point


epsilon=(2e-4);

%objective=@(f)0

objective=@(f) obj(f,n,b,SNR);

lb=zeros(n,1);
ub=b;

%opts = optimset('Display','iter','Algorithm','interior-point', 'MaxIter', 100000, 'MaxFunEvals', 100000);
options = optimoptions('fmincon','Display','iter','Algorithm','sqp');
[f,fval,exitflag,output] = fmincon(objective,x0,[],[],[],[],lb,ub,@(f)mycon_Taylor(f,SNR,n,m,bandwidth,epsilon,C_Ter,x0,b))


p=[p,-fval];

where

function fun = obj(f,n,b1,SNR)
%UNTITLED3 Summary of this function goes here
%   Detailed explanation goes here
fun=-sum((b1-f).*log(1+(SNR(1:n)./((b1-f)))))
end

and

function [c,ceq] = mycon(f,SNR,n,m,bandwidth,epsilon,C_Ter,a,b)

% Compute nonlinear inequalities at x.

c=[sum(f)- bandwidth;(1/epsilon)-sum(f(1:m)'*log(1+(SNR(n+1:n+m)./f(1:m))));
(sum(f(1:m)'*log(1+(SNR(n+1:n+m)./f(1:m))))+sum((b-a).*log(1+(SNR(1:n)./(b-a)))-(f-a).*log(1+(SNR(1:n)./(b-a)))+(SNR(1:n).*(f-a))./(b-a+SNR(1:n)))-C_Ter)];
ceq=[];

end

Actually, I deduced the initial point by solving the problem without the last non linear constraint through the following function :

function [c,ceq] = mycon(f,SNR,n,m,bandwidth,epsilon,b)

% Compute nonlinear inequalities at x.
c=[sum(f)- bandwidth;(1/epsilon)-sum(f(1:m)'*log(1+(SNR(n+1:n+m)./f(1:m))))];
ceq=[];

end

Answer 1

[Matt J]

Well, clearly the feasible set gets smaller as C_Ter gets smaller. Possibly you have made C_Ter so small that the feasible set is empty.

[wiem abd]

Dear Matt,

Thanks for your answer.

Isn't possible to find any feasible solution with these small values of C_Ter?

Actually, these are the real values of C_Ter. I cannot consider any larger values.

Is there a way to change the whole problem?

[Matt J]

I have no way of knowing for what values C_Ter will have a solution, but clearly you cannot make it arbitrarily small. For C_Ter=-inf, you will definitely have an empty feasible set. So, you will need to have a good reason to trust that any particular reduction in C_Ter from 5e+11 will be feasible.

One thing you can try is to solve your @(f)0 problem for a series of gradually descending C_Ter values:

x0=0.6*b; %Feasible for C_Ter=5e+11

for C_Ter=5*logspace(11,6,100)

 options = optimoptions('fmincon','Display','iter','Algorithm','sqp');
 [x0,~,exitflag] = fmincon(@(f) 0 ,x0,[],[],[],[],lb,ub,@(f)mycon_Taylor(f,SNR,n,m,bandwidth,epsilon,C_Ter,x0,b))
 
 if exitflag<0
     
     disp(['No solution found for C_Ter='  num2str(C_Ter)]); break

 end
 
end

Each time the solution becomes the x0 of the next for-loop iteration, so that you incrementally track the boundary of the shrinking feasible set. If the loop makes it all the way to the last iteration, you will have found a feasible x0 for C_Ter=5*1e+6. If not, there is a chance that there is no feasible point for that C_Ter.

[wiem abd]

Thank you Matt for your answer.

I chnaged the values of C_Ter as :

 C_Ter=5*10^7:5*10^7:20e+07

The problem has always a feasible point. However, I get this message

Local minimum possible. Constraints satisfied.↵↵fmincon stopped because the size of the current step is less than↵the default value of the step size tolerance and constraints are ↵satisfied to within the default value of the constraint tolerance.

When I keep changing the values of the initial point with the newly found value, I reach the following state :

Error using barrier
Nonlinear constraint function is undefined at initial point. Fmincon cannot continue.

[Matt J]

Local minimum possible. Constraints satisfied.↵↵fmincon stopped because the size of the current step is less than↵the default value of the step size tolerance and constraints are ↵satisfied to within the default value of the constraint tolerance.

This one was a good message. It probably means fmincon succeeded.