In this paper, we consider the mobile edge offloading scenario consisting of one mobile device (MD) with multiple independent tasks and various remote edge devices. In order to save energy, the user's device can offload the tasks to available access points for edge computing. Data compression is applied to reduce offloaded data size prior to wireless transmission to minimize the execution latency. The problem of jointly optimizing the task allocation decision and the data compression ratio to minimize the total tasks' execution latency and the MD's energy consumption concurrently is proposed. We show that the design problem is a non-convex optimization one but it can be transformed into a convex one through a semidefinite relaxation (SDR) based approach. Numerical simulations demonstrate the outperformance of the proposed scheme compared to the benchmark one.