Does solving a problem with less constraints leads to greater optimal value necessarily?

Dear Community,

Please find below the optimization problem I am solving using fmincon

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=500*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(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

Now I want to solve the problem with less constraints such that mycon function is given below :

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

I am expecting to get greater values for -fval using the second function (i.e. when I removed the last constraint). But, it is not always the case.

Can you help to explain this issue and what to modify in order to get the desired result?

Many thanks !

[Walter Roberson]

Not necessarily. fmincon() only looks for local minima, and a constraint potentially might act to steer it away from a local minima.

[wiem abd]

What is the reason behind better values with more constraints? I mean when I want to minimise the problem Iget -3 for example with 3 constraints and -2 with 1 constraint. I think it is absurd

Answer 1

[Walter Roberson]

Suppose you have a function that looks like

  \  /-------\     /--------
  \ /         \-v-/
   \

fmincon is deterministic, so it is likely to check out the section in the middle first. It does some local descent and finds a minima at the v . Provided the constraints are met, it is happy with that answer. fmincon is not a global minimizer: in areas where the constraints are not interfering, it follows the gradient jacobian or at most the hessian.

Now apply an additional constraint that makes the center part unfavourable. fmincon searches further and has a chance to find the region to the left that has the better minima.

When I try your code, I get

          43841811.2268468   for the first mycon
          43841811.2142779   for the second mycon with fewer constraints
  

That difference in values is small enough, 2E-10 of the value, that you could be running across tolerance issues. Or just a local minima.

[wiem abd]

Thank you for your answer Walter! It is really helpful.

Actually, C_Ter in the case of fewer constraints approaches infinity. That's why, I removed the constraint.

Is there a way to guarantee to have always the greatest values with greater C_Ter?