MATLAB: FMINCON first order optimality non-zero

fminconMATLABOptimization Toolboxsqp

Hi,
I run fmincon, sqp for a minimalization problem. The problem is setup as the IDF-architecture with a total of 34 design variables. The code exists out of multiple 'black-box' disciplines. 1 iteration takes 30 seconds.
I managed to find an optimum that meets 2 out of 3 KKT conditions. The condition it does not meet (I think) is that the gradient of the Lagrangian should be zero at the minimum. The initial point of the iteration is feasible with a first order optimality condition of 0.74. The optimum is found after 40 iterations with a total function count of around 3000 with a first order optimality condition of 0.15. The objective function decreased with 9%, which is a good result for this problem. The output message of Matlab is shown below.
However, my gut tells me that the first order optimality condition is too high, and nowhere near to zero. The optimization stopped because of the size of the current step is less than selected value, but a further decrease of this option does not lead to a lower first order optimality condition or significant changes in design vector.
How can i further investigate if this first order optimality condition is acceptable for my problem or not?
(The optimization problem has been run for multiple setups and algorithm. The options shown below gave, for the first time, an actual feasible optimized design after weeks of trying. However, is it a local minimum?)
options=optimoptions(@fmincon,...
                                  'Algorithm','sqp',...
                                  'Tolx',1e-6,...   %size smallest step. Smaller step causes optimizer to stop 
                                  'TolFun',1e-3,... %TolFun is a lower bound on the change in the value of the objective function during a step
                                  'TolCon',1e-3,...
                                  'Display','Iter',...
                                  'DiffMinChange',1e-2,...
                                  'DiffMaxChange',1e-1,...
                                  'FinDiffType','Central',...
                                  'MaxFunEvals',10000,...
                                  'MaxIter',1000,...
                                  'OutputFcn',@outputFcn_global,...
                                  'PlotFcn','optimplotfval');
    
    tic
    [x,val,exitflag,output,lambda, grad, hessian]=fmincon(@objectivefcn,x0,A,b,Aeq,beq,lb,ub,@constraints,options) 
    toc 
Local minimum possible. Constraints satisfied. fmincon stopped because the size of the current step is less than the selected value of the step size tolerance and constraints are satisfied to within the selected value of the constraint tolerance.
Exitflag 2 
output = 
 struct with fields:
 
  iterations: 40
  funcCount: 2928
  algorithm: 'sqp'
  message: 'Local minimum possible. Constraints satisfied.↵↵fmincon stopped because the size of the current step is less than↵the selected value of the step size tolerance and constraints are ↵satisfied to within the selected value of the constraint tolerance.↵↵Stopping criteria details:↵↵Optimization stopped because the relative changes in all elements of x are↵less than options.StepTolerance = 1.000000e-06, and the relative maximum constraint↵violation, 1.309327e-04, is less than options.ConstraintTolerance = 1.000000e-03.↵↵Optimization Metric                                           Options↵max(abs(delta_x./x)) =   5.25e-07                       StepTolerance =   1e-06 (selected)↵relative max(constraint violation) =   1.31e-04   ConstraintTolerance =   1e-03 (selected)'
    constrviolation: 1.309326961893564e-04
    stepsize: 1.698262377802769e-06
    lssteplength: 1.299348114471227e-06
    firstorderopt: 0.145027707751111

Best Answer

From your problem description it is very likely that the objective function is not smooth. Your black box functions may respond in nonsmooth ways to very small changes in your optimization parameter x. You may never be able to decrease your first-order optimality measure to near zero. See Optimizing a Simulation or ODE for a discussion.
You seem to have done a good job setting larger-than-default finite differences and you use central finite differences. I would have suggested these steps, but you already figured it out. As for whether you are at a true local minimum, it is hard to know. You can try looking at When the Solver Succeeds for some guidance, but I suspect that you might have already figured out these checks.
Good luck,
Alan Weiss
MATLAB mathematical toolbox documentation
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