Electricity cost is a dominant and rapidly growing expense in data centers. Unfortunately, much of the consumed energy is wasted because servers are idle for extended periods of time. We study a capacity management problem that dynamically right-sizes a data center, matching the number of active servers with the varying demand for computing capacity. We resort to a data-center optimization problem introduced by Lin, Wierman, Andrew and Thereska that, over a time horizon, minimizes a combined objective function consisting of operating cost, modeled by a sequence of convex functions, and server switching cost. All prior work addresses a continuous setting in which the number of active servers, at any time, may take a fractional value. In this paper, we investigate for the first time the discrete data-center optimization problem where the number of active servers, at any time, must be integer valued. Thereby we seek truly feasible solutions. First, we show that the offline problem can be solved in polynomial time. Second, we study the online problem and extend the algorithm {\em Lazy Capacity Provisioning\/} (LCP) by Lin et al. to the discrete setting. We prove that LCP is 3-competitive. Moreover, we show that no deterministic online algorithm can achieve a competitive ratio smaller than~3. In addition, we develop a randomized online algorithm that is 2-competitive against an oblivious adversary. Moreover, we prove that 2 is a lower bound for the competitive ratio of randomized online algorithms, so our algorithm is optimal. Finally, we address the continuous setting and give a lower bound of~2 on the best competitiveness of online algorithms. This matches an upper bound by Bansal et al. We prove that all lower bounds still hold in a problem variant with more restricted operating cost functions, introduced by Lin et al.