When optimizing a nonlinear objective, one can employ a neural network as a surrogate for the nonlinear function. However, the resulting optimization model can be time-consuming to solve globally with exact methods. As a result, local search that exploits the neural-network structure has been employed to find good solutions within a reasonable time limit. For such methods, a lower per-iteration cost is advantageous when solving larger models. The contribution of this paper is two-fold. First, we propose a gradient-based algorithm with lower per-iteration cost than existing methods. Second, we further adapt this algorithm to exploit the piecewise-linear structure of neural networks that use Rectified Linear Units (ReLUs). In line with prior research, our methods become competitive with—and then dominant over—other local search methods as the optimization models become larger.