# Heuristic artificial bee colony algorithm for solving the Homicidal Chauffeur differential game

Document Type : Original Article

Authors

1 Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran

2 Department of Electrical and Computer Engineering, SYSTEC, Faculdade de Engenharia, Universidade do Porto, 4200-465, Porto, Portugal

Abstract

In this paper, we consider the Homicidal Chauffeur (HC) problem as an interesting and practical differential game. At first, we introduce a bilevel optimal control problem (BOCP) and prove that a saddle point solution for this game exists if and only if this BOCP has an optimal solution in which the optimal value of the objective function is equal to $1$. Then, BOCP is discretized and converted to a nonlinear bilevel programming problem. Finally, an Artificial Bee Colony (ABC) algorithm is used for solving this problem, in which the lower-level problem will be considered as a constraint and solved by an NLP-solver. Finally, to demonstrate the effectiveness of the presented method, various cases of HC problem are solved and the simulation results are reported.

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