Cloud computing enables an economically promising paradigm of computation outsourcing. However, how to protect customers confidential data processed and generated during the computation is becoming the major security concern. Focusing on engineering computing and optimization tasks, this paper investigates secure outsourcing of widely applicable linear programming (LP) computations. Our mechanism design explicitly decomposes LP computation outsourcing into public LP solvers running on the cloud and private LP parameters owned by the customer. The resulting flexibility allows us to explore appropriate security/efficiency tradeoff via higher-level abstraction of LP computation than the general circuit representation. Specifically, by formulating private LP problem as a set of matrices/vectors, we develop efficient privacy-preserving problem transformation techniques, which allow customers to transform the original LP into some random one while protecting sensitive input/output information. To validate the computation result, we further explore the fundamental duality theorem of LP and derive the necessary and sufficient conditions that correct results must satisfy. Such result verification mechanism is very efficient and incurs close-to-zero additional cost on both cloud server and customers. Extensive security analysis and experiment results show the immediate practicability of our mechanism design.