This might be a very basic question for this community. I am reading an article and I think I have some confusion about formulating a problem. My understanding is that all decision variables should be present in the objective function. For any continuous function, the first derivative of the function is zero at the minima or maxima and the value of the variable for which we get the derivative equal to zero is the solution of the problem. However, the article I am reading formulated the problem as follows where some decision variables are present only in the constraints:
The decision variables are $x_a$ ,$v_\pi$, and $y_s$. However, only $x_a$ is present in the objective function. Is it because the coefficients of the absent decision variables might be $0$ in the objective function? If we modify the problem and add the constraints to the objective function like the Lagrange method we will be able to include all variables.
I would really appreciate any comments and answers.