# optimization of a separable function

this is a optimization problem whose objective function is separable:

$$F(x)=\sum_{i=1}^n\frac{\theta_i^2}{4}\sum_{j=1}^m\left(1+\rho x_ik_j+\sqrt{(x_ik_i+\rho)^2+1-\rho^2}-\frac{2\alpha_i^j}{\theta_i}\right)^2$$

where $x=(x_1,x_2,...,x_n)$ and $\rho$, $\theta_i$, $k_i$, $\alpha_i^j$ are given constants with $-1\leq\rho\leq 1$.

subject to

$$x_1\geq x_2\geq\cdots\geq x_n$$

$$\theta_1 x_1\leq\theta_2 x_2\leq\cdots\leq\theta_n x_n$$

$$0\leq x_i\leq X_i$$

where $X_i$ are also given constants.

For every component function

$$\left(1+\rho x_ik_j+\sqrt{(x_ik_i+\rho)^2+1-\rho^2}-\frac{2\alpha_i^j}{\theta_i}\right)^2$$

is not necessarily convex, we can not apply directly the Lagrangian multiplier. Another way is to solve this problem conditionally:

for $x_1,...,x_{n-1}$ fixed, consider the problem

$$F(x)=F(x_n)$$

subject to

$$x_n\leq x_{n-1}$$

$$\theta_{n-1}x_{n-1}\leq \theta_nx_n$$

$$0\leq x_n\leq X_n$$

Then we repeat by recurrence.

Because the form to optimize is not very complicated. From the computationall viewpoint or analytical viewpoint, does someone have an idea for this optimization problem?

Thanks a lot!

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