When I studied Multivariable Calculus, I discovered that each simple (one objective function and one constraint) constrained optimization problem is associated with another simple constrained optimization problem in which the roles of the objective function and constraint are swapped. For example,
A manufacturer's revenue is $100h^{2/3}s^{1/3}$, where $h$ is the number of hours of labor hired, and $s$ is the number of tons of steel purchased, and their budget is $\$20000$. Labor costs $\$20$ per hour and steel costs $\$2000$ per ton. Find the maximum revenue and the corresponding $h$, $s$, and $\lambda$, while adhering to their budget.
algebraically written as $$\begin{cases}R(h,\ s) = 100h^{2/3}s^{1/3} =\ ????\ \Tiny(objective\ function) \\ B(h,\ s) = 20h + 2000s = 20000\ \Tiny(constraint)\end{cases}$$ has the solution $R = \frac{20000\sqrt[\leftroot{-2}\uproot{3}3]{5}}{3}$, $h = \frac{2000}{3}$, $s = \frac{10}{3}$, $\lambda = \frac{\sqrt[\leftroot{-2}\uproot{3}3]{5}}{3}$. If we forget the values for $h$, $s$, $\lambda$, and the original constraint, but remember that $R = \frac{20000\sqrt[\leftroot{-2}\uproot{3}3]{5}}{3}$, we can solve
A manufacturer's revenue target is $\$\frac{20000\sqrt[\leftroot{-2}\uproot{3}3]{5}}{3}$, given by $100h^{2/3}s^{1/3}$, where $h$ is the number of hours of labor hired, and $s$ is the number of tons of steel purchased. Labor costs $\$20$ per hour and steel costs $\$2000$ per ton. Find the minimum budget and the corresponding $h$, $s$, and $\lambda$, while achieving their revenue target.
algebraically written as $$\begin{cases}R(h,\ s) = 100h^{2/3}s^{1/3} =\ \frac{20000\sqrt[\leftroot{-2}\uproot{3}3]{5}}{3}\ \Tiny(constraint) \\ B(h,\ s) = 20h + 2000s =\ ????\ \Tiny(objective\ function)\end{cases}$$ which has the solution $B = 20000$, $h = \frac{2000}{3}$, $s = \frac{10}{3}$, $\lambda = \frac{\sqrt[\leftroot{-2}\uproot{3}3]{5}}{3}$. So we've recovered the original constraint, and rediscovered $h$, $s$, and $\lambda$. This provides an interesting, higher-level perspective into constrained optimization. We can now see that it is somewhat arbitrary which is the constraint, and which is the objective function; the choice depends only on what information is known about the real-world problem.
Recently, I found myself solving for the Nash Equilibrium quantities produced by firms in Cournot Duopolies and Stackelberg Duopolies. As I understand, solving a Stackelberg Duopoly for the first firm is a constrained optimization problem in which the constraint is derived from the second firm's optimization problem. There is a familiar vibe here of swapping the objective function and constraint, this time by changing the order in which the two firms make decisions.
Thus I hypothesized that Cournot/Stackelberg duopoly problems might be a fleshing out of the above generalization of constrained optimization. However, in the former case, the answer to the associated constrained optimization problem is the same as the original, whereas changing the order of decisions in Stackelberg competition can actually change the Nash Equilibrium quantities. Are these two types of actions that swap an objective function with a constraint related? What is the general way to think about swapping objective function and constraint that covers both types of actions?