We consider variational problem of finding the cost-optimal path on the surface of the terrain. In our model the path is represented as a curve on a plain, while the actual terrain is taken into consideration by the cost function. Our main goal is to obtain an approximate solution as a piecewise linear function via simulated annealing algorithm. For this purpose, we introduce a uniform grid and solve a problem of finding least-cost path with transition prices obtained as the values of the integral cost functional. We adapt the simulated annealing algorithm for this problem and compare its performance with a set of other solutions, namely modified A* and ant colony optimization algorithm. To obtain a better solution we use modifications of the algorithm, such as quantum annealing and stochastic tunneling, which help us improve the performance and avoid getting stuck in the local optima. We also compare the used approaches and provide numerical examples of application of the introduced methods.