# Optimization problem where convex and non-convex problems are “ mixed ”

I have a naive question for an optimization problem. Let $$F(x_1,x_2,x_3)=(a-x_1 \frac{(1-x_2)\sin x_3}{1-x_2 \cos x_3} )^2 + (b-x_1 (1-x_2^2))^2$$ where $a$ and $b$ are known values, I have an optimization problem that consists in minimizing $\min F (x_1,x_2,x_3)$ under the constraints $$x1 \geq 0, 0\leq x_2 \leq 1, 0\leq x_3 \leq \pi/4$$

I find that $F(x_1,x_2,x_3)$ is convex w.r.t. variable $x_1$, but is not to $x_2$ or $x_3$. My question is, is it reasonable to do the optimization in two separate steps: first fix $x_1$, minimize $x_2$ and $x_3$, and second fix $x_2$ and $x_3$ to minimize $x_1$? In other words:

1. Initialization of $x_2$ and $x_3$
2. Fix $x_2$ and $x_3$, update $x_1$ using linear least square method.
3. Fix $x_1$, update $x_2$ and $x_3$ using interior-point algorithm.
4. Stop iteration if convergence. Otherwise, go to #2.

Is this a reasonable process to separate the minimization problem to convex and non-convex, and solve them in stages?

I also find that there are a lot of discussions on how to identify convex problem, how to transfer problems to convex problems, but I cannot find information about how to deal with the case where convex and non-convex problems are " mixed ".

Any ideas? Thank you in advance.

• How accurate does your solution need to be? Since you only have three unknowns, a brute force grid search could be an easy option. – littleO Nov 18 '13 at 9:59
• @littleO I doubt grid search would be appropriate in many cases. Although there are only 3 unknown for a single optimization problem instance, there might be tens of thousands of instances. Grid searching should be too slow for that purpose. – zell Nov 18 '13 at 16:15