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Numerical stability (computations going sideways) and numerical tolerances are related but not identical. Floating point arithmetic being subject to rounding and truncation errors (unavoidably), every solver will need to treat things that are "nearly nonnegative", "nearly zero" or "nearly integer" as if they are in fact nonnegative / zero / integer. That ...


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(1) Numerical stability is a real issue but not so common in my area of discrete optimisation (for obvious reasons). I only get it in two cases. Sometimes I get preprocessed data that has been rounded, resulting in nearly parallel constraints. Other times I badly implement an iterative algorithm like Benders decomposition, which produces more numerical error ...


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The comment by @prubin is spot on. The V matrix is probably being numerically evaluated as not positive semidefinite by CPLEX. That can easily happen when one ore more eigenvalues are what you call null, i.e., theoretically equal to zero, i.e., zero in exact arithmetic. But CPLEX is using double precision floating point arithmetic, not exact arithmetic, so ...


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Cplex can minimize convex quadratic objectives, or maximize concave ones. For example, minimizing $x^2+y^2$ is OK, but minimizing $x^2 -y^2$ is not. See this page for more details: As mentioned in the page, you can set the parameter optimalitytarget to 2 to proceed and accept the risk of finding a local optimum. If so, Cplex will not stop and look for an ...


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