In this work, we define the generalized wake-up problem, $GWU(s)$, for a shared memory asynchronous system with $n$ processes. Informally, the problem, which is parametrized by an increasing sequence $s = s_1,\ldots,s_p$, asks that at least $n - i + 1$ processes identify that at least $s_i$ other processes have "woken up" and taken at least one step for each $1 \le i \le n$. We prove that any solution to $GWU(s)$ that uses read/write/compare-and-swap variables requires at least $Ω\left(\sum_{i = 1}^n \log s_i \right)$ steps to solve. The generalized wake-up lower bound serves as a technique for proving lower bounds on the amortized complexities of operations on many linearizable concurrent data types through reductions. We illustrate this with several examples: (1) We show an $Ω(\log n)$ amortized lower bound on the complexity of implementing counters and {\em fetch-and-increment} objects which match the complexities of the algorithms given by Jayanti and Ellen & Woelfel; the lower bound even extends to a significantly relaxed version of the object. (2) We show an $Ω(\log n)$ amortized lower bound on the complexity of the pop, dequeue, and deleteMin operations of a concurrent stack, queue, and priority queue respectively that hold even if the data type definitions are significantly relaxed; (3) In another paper, we have shown an $Ω(\log\log(n \ell/m))$ amortized lower bound on the complexity of operations on a union-find object of size $\ell$ (when $m$ operations are performed).