Fmincon linear contraint doens't work in my case

Hi there,

I am solving a nonlinear optimization problem. The code shows as follows in which I have 54 unknowns.

A = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0;
       0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0;
       0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0;
       0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0;
       0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0;
       0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0;
        0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
      0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1];
b = [0.015;-0.005;0.015;-0.005;0.08;-0.02;0.08;-0.02];
% 0.005<=x(51)<=0.015;
% 0.005<=x(52)<=0.015;
% 0.02<=x(53)<=0.08;
% 0.02<=x(54)<=0.08;
Aeq = [];
beq = [];
x0=  0.001*rand(54,1);
lb= -200*ones(54,1);
ub= +200*ones(54,1);
options = optimoptions('fmincon','Display','iter','MaxFunctionEvaluations', 10000,'ConstraintTolerance',1e-10);
[x,y,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN]=fmincon('fun1',x0,A,b,Aeq,beq,lb,ub,'fun3',options);

fun1：
f=(x(49))^2;
fun3：
(I have got a lot of nonlinear and linear equations here and some inequality contraints here. Altogether around 80 constraints. 
Thoes equations have no bugs or errors as they can run successfully under fsolve function if the 4 unknowns are given.)

The idea is to optimize the last four unknowns within their ranges shown above(A,b).

Sometimes, fmincon can give some 'good' results saying local minimal value found. However, when I trace back to the values of the last 4 unknowns, there are not actually within the ranges set in the code.

I was wondering if anyone could help me with this?

My email address is [e-mail]. Contact me if you want to look into the case.

Thanks in advance,

Ke

[Walter Roberson]

Is there a reason you did not code that A b in terms of lower bound and upper bound parameters?

[Wu Ke]

I tried this as well, but didn't work out. The same results got as the results shown above.

Answer 1

[Matt J]

You should check the EXITFLAG output to see if fmincon believes it converged. If it did converge, the ranges should be satisfied to within the ConstraintTolerance parameter that you set.

As Walter alluded, it would really be better for you to express the desired ranges using the lb, ub inputs. That is their purpose. Moreover, under default settings, the lb, ub constraints will be ideally satisfied (no violation of the bounds is tolerated).

[Walter Roberson]

A couple of days ago someone happened to post a ga task with problem based optimization in which they were running out of memory, and when I tracked it turned out that the construction of the solver optimization form from the problem based form for ga involved converting lb and ub into entries in the A / b matrices. The user had over 100000 variables and you need a diagonal matrix to express all of those lb entries through the A matrix, so they needed a matrix with over 100 gigabytes of main memory just to handle the lb .

Yes, in theory the two forms are equivalent, but the amount of extra memory used up was atrocious :(

[Wu Ke]

I tried lb,ub as well, but didn't work out. The same results got as the results shown above.

I also checked the exitflag which was positive something like 2 or 3. This could be considered as 'correct'.

Besides, MaxFunctionEvaluations and ConstraintTolerance were both taken into account.

It can find the 'optimized value' sometimes but it is most likely to loop forever under

if (EXITFLAG==1)||(EXITFLAG==2)||(EXITFLAG==3)||(EXITFLAG==4)
break
end

‘Converged to an infeasible point’ is the most results I got.

[Matt J]

The same results got as the results shown above.

Shown where? I don't see any.

I also notice that you are making a very peculiar initial guess

x0=  0.001*rand(54,1);

This initial guess doesn't even satisfy your bounds. Although it is not strictly necessary that they do, I don't know why you would initialize the iterations farther from the solution than you have to. Also, you seem to say in your post that fsolve successfully finds a solution to your nonlinear equality constraints. If you can choose x0 to satisfy the nonlinear constraints, that would be helpful to fmincon as well.

In any case, it would be helpful for us to have fun3 so we could test the code ourselves.

[Wu Ke]

Sorry. I didn't tell it right. I meant trying lb and ub does not make a difference compared to the results of using A b inequality constraints.

For 'This initial guess doesn't even satisfy your bounds. ', see below.

x0=  0.001*rand(54,1);
lb= -200*ones(54,1);
ub= +200*ones(54,1);

I don't understand the initial guess x0 does satisify my bounds. rand returns random values within the range of (0,1). It should satisfy my lb and ub.

Oh sorry! I got it. xo doesn't satisfy my A b inequality contraints. I will take a look.

For 'initialize the iterations farther from the solution than you have to':

This is because it kept saying 'fmincon stopped because it exceeded the function evaluation limit' without optimizing further. I thought this might be the reason of not converging so I increased it.

See the code attached.

Thanks,

Ke

[Matt J]

A few notes on fun3.

(1) You have linear equality constraints implemented in fun3. The appropriate place for those is the Aeq,beq matrices. Similar to what we've been saying about bounds, you make it harder for fmincon when you hide the simplicity of certain constraints by putting them in the wrong place.

(2) Your nonlinear inequality constraints can be reformulated as linear and therefore moved out of fun3 and into A,b. For example, because x(52)>0, we can re-arrange this inequality

 x(1)/x(52)-0.1<=0

into this linear form,

 x(1)-0.1*x(52)<=0

[Wu Ke]

I have made the changes you mentioned but it's still not working saying:

Converged to an infeasible point.

Do you know any other plugin for MATLAB that is capable of solving this sort of problems?

[Matt J]

If we're not sure there is a feasible point, there really isn't anything to recommend. You mentioned above that you were successfully solving this with exitflag=2 or 3. What happens when you initialize with those solutions?

[Wu Ke]

I did get some results exitflag with exitflag=2 or 3. The problem is the 4 last unknowns analysed are not in the ranges set in the code. Apart from the 4 unknowns, the other 50 unknowns also don't make sense in physics (eventhough I tried setting up some reasonable ranges for them, the solver didn't find the feasible points) as the whole mathematical model is based on a physical model.

[Matt J]

It is not possible that you got exitflag=2 or 3 but violated the bounds. I suggest you post your latest code and also the x0 that leads to exitflag 2 or 3 so that we can try to reproduce it.

[Wu Ke]

Will do!

I really appreciate it!

Ke