# Second-Order Cone Program?

 0 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 second-order 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 second-order 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 15●3 accept rate: 0%

 0 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 non-convex constraints. Counter-example which shows this is non-convex: Point A = (0,0,0) and point B = (1,1,-0.5) both satisfy the constraints. The convex combination, C = 0.5*(A + B), does not satisfy the constraints. answered 22 Oct '17, 15:26 Mark L Stone 447●3●11 accept rate: 15%
 0 Hint: complete the square for $$x_3$$. answered 16 Oct '17, 14:26 Rob Pratt 1.2k●2●6 accept rate: 28% 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
 toggle preview community wiki

By Email:

Markdown Basics

• *italic* or _italic_
• **bold** or __bold__
• image?![alt text](/path/img.jpg "Title")
• numbered list: 1. Foo 2. Bar
• to add a line break simply add two spaces to where you would like the new line to be.
• basic HTML tags are also supported

Tags: