Optimal search for a moving target with no time information maximizing the expected reward

This paper investigates a search problem for a moving target on a network in which any time information of the target position is not available to a searcher. The searcher has to distribute the limited amount of search efforts on a search space to detect the target, knowing only route information of target paths but not time information about when the target passes there. On detection of the target, the searcher gains some value but expends search cost. There have been few papers which mathematically deal with such a search model without any time information of the target position so far. We formulate the search-efforts-optimizing problem under the expected reward criterion as a convex programming problem and obtain necessary and sufficient conditions for optimal solutions. Using the conditions, a new algorithm is proposed to give an optimal solution. It is shown that the algorithm has the high efficiency for computational time and the robustness for the size of problems comparing with some well-known methods for non-linear programming problems: the gradient projection method and the multiplier method, by numerical examinations. We also elucidate some properties of the optimal solution by the sensitivity analysis of system parameters.