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Resource allocation for Device-to-Device (D2D) communication underlaying cellular network poses new challenges in terms of interference while at the same time provides increased system sum rate. In this paper, we propose a local search based resource allocation algorithm (LORA) for allocating resource blocks to D2D devices that are shared with Long Term Evolution (LTE) cellular users. We first formulate the problem of downlink resource block (RB) allocation to D2D users from cellular users as a computationally expensive mixed integer nonlinear programming (MINLP) problem. However, as the optimal solution of an MINLP can take exponential time to compute, we propose a local search based algorithm to compute a locally optimal solution based on an initial feasible solution. We compare the obtained system sum rate from this local search algorithm with a well-known greedy heuristic based resource allocation algorithm and a random resource allocation algorithm. The simulation results show that LORA achieves an overall better system sum rate compared to the other algorithms for RB allocation while maintaining the signal quality at the cellular users and the D2D receivers.