Assume every linear operation (such as inner product with constant vector) can be performed in one step and multiplication by variables (quadratic operation) can be performed in one step.
We know the straight line complexity of $$(\sum_{i=1}^kx_i)^2$$ is linear in $k$ basic model and $2$ in this new modified model (adding one step and squaring one).
What is the modified complexity of $$(\sum_{i=1}^k\alpha_ix_i^2)+\sum_{i=1}^k\beta_ix_i(\sum_{\substack{{j=1}\\ i\neq j}}^kx_j)$$ where $\alpha_i>\frac{\beta_i^2}2+\beta_i$ holds at every $i\in\{1,\dots,k\}$?
I am just wondering if it is $o(k)$ at some special $\alpha_i$,$\beta_i$ with $\alpha_i>\frac{\beta_i^2}2+\beta_i$?
I am NOT reasonably sure the complexity is similar to $\alpha_i<\frac{\beta_i^2}2+\beta_i$.
