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(eng): In this paper, we propose a new numerical method to compute approximate and the so-called relaxed pessimistic solutions to mathematical programs with equilibrium constraints (MPECs), where the solution map arising in the equilibrium constraints is not single-valued. This method combines two types of existing codes, a code for derivative-free optimization under box constraints, BFO or BOBYQA, and a method for solving special parametric MPECs from the interactive system UFO. We report on numerical performance in several small-dimensional test problems.