fmincon is nonlinear constraints satisfied, then not satisfied when I include objective function

My problem is that I'm trying to optimise something with fmincon, and it can't find a point which doesn't violate the non-linear constraints. The strange thing is that it can find such a point if the objective function is replaced with a dummy function - so I know that the non-linear constraints themselves aren't the problem. Here's the structure of my code:

%First some set up
args = MakeArgs; %These are just arguments used for the objective and constraints
bounds = MakeBounds(args);
LC = MakeLinearConstraints(args);
x01 = ones(1,args.xlength); %This and f are dummy vectors
f = zeros(1,args.xlength);

optionsLinProg = optimoptions('linprog','Display','off');

%Next check that the linear constraints are OK
[xlin , ~, linflag] = linprog(f,LC.A,LC.B,LC.Aeq,LC.Beq,bounds.LB,bounds.UB,optionsLinProg);
%Then a potential error warning
if ~(linflag==1)
    error('Error. The linear constraints are badly defined')
end
fprintf('Linear Satisfied');
xlin = xlin'; %This is because fmincon deals with row vectors, wheras linprog uses column vectors.

options = optimoptions('fmincon');
options = optimoptions(options,'Algorithm','sqp');

%Next we want to test the nonlinear constraints. Again, we create a dummy function (zf) to serve as the objective. 
nlconopt = @(y) nlcon(y,args,x01);
zf = @(x)0;
%
[xnlin,~,nonlinflag,~] = fmincon(zf,xlin,LC.A,LC.B,LC.Aeq,LC.Beq,bounds.LB,bounds.UB,nlconopt,options);
% Next we have a potential error warning
if ~(nonlinflag == 1)
    error('Error. Non linear constraints not satisfiable here.')
end
%And a success output
fprintf('Nonlinear constraints satisfied\n')

Now, up until this point, everything seems to work fine. The "Nonlinear constraints satisfied" message outputs every time, with various choices going into the args.

But when I then replace the dummy objective zf with the real one, and use xnlin as a starting point (The following code comes directly after the code above):

objectiveopt = @(y) objective(y,args,x01);

[xnew,newval,xnewflag,out] = fmincon(objectiveopt,xnlin,LC.A,LC.B,LC.Aeq,LC.Beq,bounds.LB,bounds.UB,nlconopt,options);
if ~(xnewflag == 1)
    error('Error. Non linear constraints not satisfied when taken with objective.')
end

I get my error message every time. This initially because it hits the evaluations limit - if I increase this, then it hits the iterations limit. I can play with these but what perplexes me is that it should fail to satisfy the constraints at all, or converge to an infeasible point. Since the bounds and constraints are all clearly satisfiable (as shown by the code up until this point), and since it is starting from an initial point which satisfies all of them, I can't understand how the optimisation is crashing out in this way.

I've also tried it with active-set and interior-point algorithms, to no avail.

Answer 1

[Matt J]

The algorithms of fmincon try only to satisfy the nonlinear and A,b,Aeq,beq constraints at convergence. They do not pledge to satisfy those constraints at all iterations, nor to remain in the feasible region even if they are lucky enough to land there at some iteration. Even the lb,ub bounds may not be satisfied under certain non-default options settings.

Since in your case fmincon bails out early because of various tolerances and stopping limits, convergence does not take place and the nonlinear constraints never get satisfied.

[Dan Goldwater]

Thanks, I think this is possibly the heart of my problem.

Is there a minimizer which will always return a solution which respects the constraints? It seems strange that the convergence is prioritised over the constraints.

[Matt J]

Is there a minimizer which will always return a solution which respects the constraints?

No, but how would it help you if the constraints were respected? Since your optimization is quitting early, the result returned by fmincon is undoubtedly non-optimal. Does it do you any good to have a returned value that satisfies the constraints, but is non-optimal?

It seems strange that the convergence is prioritised over the constraints.

I would need you to elaborate on why that seems strange to you. Are your nonlinear-constraints convex? For general, non-convex constraints, there is no guarantee that a monotonically descending path exists from a feasible starting point to the (global) minimum. So, it's best to allow your search to consider less restricted paths.

Also, with nonlinearly constrained feasible sets, it is much more difficult to stay in the feasible set once you get there than with a linearly constrained one. With linear constraints, there are systematic ways to calculate a local set of nearby feasible points to step to for the next iteration, but not so with non-linear constraints.

[Dan Goldwater]

In reply to your question 'No, but how would it help you if the constraints were respected?' - The thing is that I need to get values out for this function, and they must respect all constraints. Ideally they'd also be global minima, but if they're sub-optimal that's something I can manage. What I can't be doing is getting results which violate the constraints or lie outside the bounds - I can't use those.

As to why it seems strange to me - I suppose I'm not empathising enough with how fmincon finds the solutions. It's just that on the face of it, I'd rather have legal solutions than optimal, illegal ones.

I'm not sure if my problem is convex. I suspect it might not be (I have six non-linear inequalities and one non-linear equality, and one of the inequalities there involves an arc-cos function). To be honest I'm not sure how I'd check if it were convex.

[Matt J]

Ideally they'd also be global minima, but if they're sub-optimal that's something I can manage.

In that case, the solution seems simple: if xnew from the final optimization step doesn't satisfy the constraints, then return xnlin instead, which you know does satisfy them.

Another possibility to consider is that your tolerances may be too tight, and that that is why fmincon is reaching maximum iteration limits.

[Nourhan Elsayed]

Hi Matt J

i am facing a similar problem, where my priority is to extract the points that satisfy the nonlinear equality constraint. whould you please clarify what is xnlin and how to return that ?

Thank YOu

[Matt J]

@Nourhan, xnlin was generated in the OP's code on this line

[xnlin,~,nonlinflag,~] = fmincon(zf,xlin,LC.A,LC.B,LC.Aeq,LC.Beq,bounds.LB,bounds.UB,nlconopt,options);