I have this optimization problem: $$x_{1} \cdot x_{2} \geq (x_3)^2$$ where \(x_1, x_2 \geq 0\) I know this is a rotated secondorder cone, which can be transformed into a cone using this formulation: $$\lVert [x_{1}  x_{2},\ 2\cdot x_3] \rVert \leq x_{1} + x_{2}$$ However, as I expand my optimization model, I added a linear term to my constraint: $$x_{1} \cdot x_{2} \geq (x_3)^2  x_3$$ I have tried to express this as a secondorder cone but I am pretty sure it is not possible. Could you help me identify which type of constraint this is, and which type of optimization program I could use to solve it? (the rest of the constraints in my model and the objective function are linear) asked 16 Oct '17, 14:15 Luis Badesa 
With the linear term, it is not a Second Order Cone Program, because it is not convex. So you need a solver which can handle nonconvex constraints. Counterexample which shows this is nonconvex: answered 22 Oct '17, 15:26 Mark L Stone 
Hint: complete the square for \(x_3\). answered 16 Oct '17, 14:26 Rob Pratt Hi @Rob, I tried that before, which gives: $$x_1 \cdot x_2 \geq (x_3\frac{1}{2})^2\frac{1}{4}$$ However, I don't see how it's possible to express this as an SOC. I have tried things like: $$\lVert [x_1  x_2  \frac{1}{2},\ 2\cdot (x_3\frac{1}{2})] \rVert \leq x_{1} + x_{2} + \frac{1}{2}$$ But I don't see any combination of signs in the \(\pm x_{1} \pm x_{2} \pm \frac{1}{2}\) terms that would keep the \((\frac{1}{2})^2\) and \(x_1\cdot x_2\) terms... Is there any other way of expressing the SOC?
(16 Oct '17, 16:52)
Luis Badesa
