I need to optimize sum $\sum_{i,j} \alpha_i \beta_j u_{i,j} \to \min$, where $\alpha_i \in [0,1]$ and $\beta_j \in [0,1]$ are variable and $u_{i,j} \in \mathbb{R}$ are given.
In other words, I need to find vectors $\alpha,\ \beta$ with values in $[0,1]$ such that the weighted matrix sum is minimal.
The sum can be rewritten as
- $\alpha\ U \beta^T \to \min$,
- $\sum_{i,j} (\alpha^T\beta )_{i,j}\ \cdot\ u_{i,j} \to \min,$ (*)
- $\text{trace}(\beta^T\alpha\ U) \to \min.$
Are there ways to solve this other than optimize numerically? At first I thought it can be solved using linear programming, for example for the case (*), but I need to find two vectors, not one.