Suppose we have the following objective function with one decision variable $x_i$ where $p_i$ is a fixed parameter for each $i$ and also, $a$ is a constant for the problem
\begin{align}
\label{eq} \max \sum^N_{i=1} p_i(a+1-x_i) = \max \bigg{(}\sum^N_{i=1} p_ia +\sum^N_{i=1}p_i -\sum^N_{i=1} p_i x_i\bigg{)}
\end{align}
As the first two components: $\sum^N_{i=1} p_ia $ and $\sum^N_{i=1}p_i$ are fixed and changing $x_i $ won't affect them anyway. Is there any difference if I excluded the fixed components from the objective function and set the objective function to be \begin{equation} \max \sum^N_{i=1} -p_ix_i = \min \sum^N_{i=1} p_ix_i? \end{equation}
Of course, the objective function value is not the same before and after excluding the fixed components, but will the values of $x_i$ be the same in both cases?
