Source localization within wireless sensor networks (WSNs) is one of the critical technologies in the Internet of Things (IoT). As the number of network nodes increases, so does the amount of data and computational requirement. It is imperative to introduce edge computing. However, there are still two issues when running existing wireless location algorithms on edge nodes: 1) conventional low-complexity approaches are easily affected by the bias generated in complex environments, leading to low locating accuracy and 2) the optimization algorithms considering the bias have good performances, but they are calculation-efficiency low on edge nodes. This study proposes a computationally efficient and high-precision location method to tackle the troubles. Precisely, we first introduce our previous research to construct a bias-considered nonconvex problem with a linear objective. Then, we propose an angle-assisted Taylor series with zero truncation error to linearize the second-order cone (SOC) constraint in the established problem. Next, we resort to the mini-max criterion to eliminate the angular uncertainty and get a robust linear programming (LP) problem with an optimal solution. So far, we have obtained a convex problem of low complexity. To ensure the calculated efficiency of the proposed problem on edge nodes, we proceed to give the solving process of the problem. Moreover, we provide a constraints tracking mechanism to reduce the number of iterations in the solution procedure, improving computational efficiency. Simulations and experiments demonstrate that the proposed method with similar locating accuracy to state-of-the-art optimization algorithms exhibits much higher computing efficiency on edge nodes.