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In this paper, we investigate the problem of optimal content cache management for HTTP adaptive bit rate (ABR) streaming over wireless networks. Specifically, in the media cloud, each content is transcoded into a set of media files with diverse playback rates, and appropriate files will be dynamically chosen in response to channel conditions and screen forms. Our design objective is to maximize the quality of experience (QoE) of an individual content for the end users, under a limited storage budget. Deriving a logarithmic QoE model from our experimental results, we formulate the individual content cache management for HTTP ABR streaming over wireless network as a constrained convex optimization problem. We adopt a two-step process to solve the snapshot problem. First, using the Lagrange multiplier method, we obtain the numerical solution of the set of playback rates for a fixed number of cache copies and characterize the optimal solution analytically. Our investigation reveals a fundamental phase change in the optimal solution as the number of cached files increases. Second, we develop three alternative search algorithms to find the optimal number of cached files, and compare their scalability under average and worst complexity metrics. Our numerical results suggest that, under optimal cache schemes, the maximum QoE measurement, i.e., mean-opinion-score (MOS), is a concave function of the allowable storage size. Our cache management can provide high expected QoE with low complexity, shedding light on the design of HTTP ABR streaming services over wireless networks.