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In economics, one often face the following maximization problem: $ \max_{{\{c(t)\}}_{t=1}^\infty}\sum_{t=0}^\infty \beta^tu(c(t)), $ where $\beta$ is the discount factor, $u(c(t))$ is the utility of consumption at time $t$. Very often we have one constraint for each time period in this type of problems, so there end up being infinite number of constraints. Many just proceed to take FOCs as if there were finite number of constraints. My question is what theorem guarantees that Lagrange's method work for infinite number of constraints? What about Kuhn-Tucker conditions?

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To my knowledge there is no theorem allowing to use the Kuhn-Tucker conditions with an infinite number of constraints. However, if these constraints are non-negativity constraints (or other simple restriction on the domain of the solutions), and if the problem is convex (or concave), one can use a generalization of Karush-Kuhn-Tucker theorem to series, from Bachir & Fabre (forthcoming), see this other post.

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