Quadprog message: The problem is non-convex
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First of all, I am using Matlab 2017b and the optimalization toolbox of 2019. I am using quadprog (not suprisingly ;)) to solve a QP problem. In reality, my problem is convex, however I get in the command window a message from quadprog that my problem is non-convex. I am not using lower and upper bounds, but inequality constraints. Furthermore, I checked if I had negative eigenvalues in my Hessian. This was indeed the case. I had one or two really small negative eigenvalues (-1e20). However, I am using quadprog for multiple timesteps, whereby the lineair term f (in my case F) is updated and therefore changes a little bit after every time step. If I am running the code for example for 10 timesteps, I get 8 times the message that the problem is non-convex and two times that the problem is convex and a minimum has been found. This is in my opinion quite weird, because the Hessian stays the same every timestep (and with that the one or two really small negative eigenvalues).
Does anyone have a potential solution to fix this issue or the cause of it? If you think you need more information, please let me know.
[X,fval,exitflag,output,lambda] = quadprog(G,F,Aineq_SparseCondensed,bineq_SparseCondesed,,,,,,options);
G = (G+G')/2;
Thanks in advance!
Matt J on 23 Jan 2020
Edited: Matt J on 23 Jan 2020
You cannot submit a problem that is borderline convex to quadprog if it is using an algorithm that expects convexity. If you do, quadprog can see the problem as convex or not-convex arbitrarily, depending on floating point numerical errors that essentially contribute random noise to the calculations that it is doing.
I would guess that your Hessian is supposed to be more distinctly positive definite, so you should review the way you set up the problem.