Logarithmic water filling ($x_i=(\tau -{\alpha }_i{)}_{+},{\sum }_i{x}_i=B$)

To maximize ${\sum }_i\log ({\alpha }_i+x_i)$ for positive baselines, nonnegative allocations and a positive budget $B$, the KKT conditions equalize the shifted values on allocated coordinates. The unique water level solves ${\sum }_i(\tau -{\alpha }_i{)}_{+}=B$. Inactive coordinates have baselines at least the water level. Strict convexity of the negative objective ensures uniqueness, and sorting the baselines gives an efficient water-filling algorithm.