Dear all,

I am struggling with quadratic optimization right now. I have a problem with a linear objective and several linear constraints. However, one set of constraints takes the form $c \geq a / (a+b)$ which obviously is a quadratic constraint. Here, $a,b$ are non-negative real numbers, while $c$ takes any value between $0$ and $1$. On the selected domain, the quadratic matrix is positive semi-definite, but not on the entire area (so if $a,b,c$ take negative values). 

Is it possible to still work with this constraint in CPLEX?
If push comes to shove, I will replace this with a few piece-wise linear functions, but I was hoping that there was a solution without any binary/integer variables. 

Thank you!
#DecisionOptimization

Presumably your first step is to convert the inequality to $ac + bc - a \ge 0$, which is quadratic. I'm not sure what you mean by the quadratic matrix being positive semidefinite "on the selected domain", since the Hessian matrix is a constant (and is not psd). In any case, the feasible region is clearly not convex. For instance, (a, b, c) = (0, 0.1, 0) and (a, b, c) = (10, 0, 1) both satisfy the constraint but their midpoint (5, 0.05, 0.5) does not. CPLEX can solve nonconvex QPs and MIQPs (I think), but as far as I know it cannot solve nonconvex MIQCPs (i.e., where the feasible region is nonconvex). So I think you are going to have to do a PWL approximation or perhaps switch to a nonconvex solver.

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Paul Rubin
Professor Emeritus
Michigan State University
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