optimization of a separable function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:44:03Z http://mathoverflow.net/feeds/question/102454 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102454/optimization-of-a-separable-function optimization of a separable function Higgs88 2012-07-17T15:01:04Z 2012-07-31T12:33:42Z <p>Hello everyone,</p> <p>this is a optimization problem whose objective function is separable:</p> <p>$$F(x)=\sum_{i=1}^n\frac{\theta_i^2}{4}\sum_{j=1}^m\left(1+\rho x_ik_j+\sqrt{(x_ik_j+\rho)^2+1-\rho^2}-\frac{2\alpha_i^j}{\theta_i}\right)^2$$</p> <p>where $x=(x_1,x_2,...,x_n)$ and $\rho$, $\theta_i$, $k_i$, $\alpha_i^j$ are given constants with $-1\leq\rho\leq 1$.</p> <p>subject to </p> <p>$$x_1\geq x_2\geq\cdots\geq x_n$$</p> <p>$$\theta_1 x_1\leq\theta_2 x_2\leq\cdots\leq\theta_n x_n$$</p> <p>$$0\leq x_i\leq X_i$$</p> <p>where $X_i$ are also given constants.</p> <p>For every component function</p> <p>$$\left(1+\rho x_ik_j+\sqrt{(x_ik_j+\rho)^2+1-\rho^2}-\frac{2\alpha_i^j}{\theta_i}\right)^2$$</p> <p>is not necessarily convex, we can not apply directly the Lagrangian multiplier. Another way is to solve this problem conditionally:</p> <p>for $x_1,...,x_{n-1}$ fixed, consider the problem </p> <p>$$F(x)=F(x_n)$$</p> <p>subject to</p> <p>$$x_n\leq x_{n-1}$$</p> <p>$$\theta_{n-1}x_{n-1}\leq \theta_nx_n$$</p> <p>$$0\leq x_n\leq X_n$$</p> <p>Then we repeat by recurrence.</p> <p>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?</p> <p>Thanks a lot! </p>