Given tall matrices $A, B \in \mathbb{R}^{n \times k}$, where $n \gg k$, let $C := AB^T$. I want to calculate $$ \max C = \max AB^T$$ where $\max: \mathbb{R}^{n\times n} \rightarrow \mathbb{R}$ returns the maximal entry in a matrix. Is there a way to calculate the max element without explicitly calculating $C$? This is beneficial in case $n$ is large.
Background
This problem arises when solving LASSO for large scale data. Basically you choose an active set (i.e., parameters that are not zero and then you set the others to zero) which is relatively small to the number of parameters you want to optimize over. You optimize over the active set using whatever algorithm and then you check the optimality condition for the zero coefficients, which is defined as the maximal element in the multiplication of the residual matrix with the data matrix is less than the regularization parameter. Multiplication the residual (i.e., $A$) with the data (i.e., $B$) is the computational bottleneck.