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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?

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  • $\begingroup$ 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. $\endgroup$ – David M. Apr 1 at 0:33
  • $\begingroup$ Thanks, @DavidM. $\endgroup$ – Soodi Apr 1 at 12:36

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