# Optimization problem with the dependent decision variables

In the optimization problem with $$2$$ dependent decision variables, is it possible to reduce the objective function to a problem with one variable and enforce the relation between variables as an equality constraint?

For example, suppose we aim to maximize an objective function $$f(q_1,q_2)$$ with two correlated variables $$q_1$$ and $$q_2$$ (such as weight and size) and their relation is: $$q_1=g(q_2)$$. Is it correct to keep $$q_1$$ in the objective function and have "maximize $$f(q_1)$$" and add "$$q_1=g(q_2)$$" as a constraint? Is it correct to have such constraint, theoretically?

• I think you're mixing up $q_1$ and $q_2$ a little but the idea seems fine (theoretically). If $q_1=g(q_2)$, then your objective function would be $f(g(q_2),q_2)$ (i.e. you eliminated the $q_1$ variable). These problems are mathematically equivalent. In terms of solving the problem numerically, there may be times to prefer one formulation over the other. – David M. Apr 1 at 0:33
• Thanks, @DavidM. – Soodi Apr 1 at 12:36