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I have linearly constrained QP problems with multiple disjoint global optima (not PSD). I'd like to make a QP solver characterize all of the globally optimal solutions. Here is a very simple example of the sort of case that arises:

minimize f(x,y)=x*y
subject to:
[1] (x + y) <= 1
[2] (x + y) >= 0.5
[3] x >= 0
[4] y >= 0

Here, there is a continuous segment of optimal (f=0) solutions on the x-axis [0.5,1], and another disconnected segment of optimal solutions on the y-axis. I've posed this problem to BARON with high values of NumLoc and NumSol, but it seems to give up prematurely after finding a single point solution. (Tried BARON since it does seem adept at finding multiple point solutions in other problems.)

Are there better methods or solvers for such problems?

Rodrigo de Azevedo
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S. Farlies
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  • You can reformulate the KKT conditions as a linear MIP ([link](http://yetanothermathprogrammingconsultant.blogspot.com/2016/06/solving-non-convex-qp-problems-as-mip.html)). Then standard methods (solution pool, integer cuts) can be used to find alternate solutions. – Erwin Kalvelagen Jun 27 '18 at 13:39
  • The MIP translation suggested by Erwin Kalvelagen is looking promising; however it is a bit of work. I had hoped one of the global solvers (tried BARON and ANTIGONE) would have a robust answer to this type of problem, i.e. a certificate that all branches were explored, but possibly this is too much to hope. – S. Farlies Jun 28 '18 at 19:44

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