An algorithm is proposed to solve multiextremal problems with nonconvex constraints. The algorithm uses the values of the first derivatives of the problem which are assumed to satisfy the Lipschitz condition. An index scheme is used to reduce the constrained problem to the unconstrained one. The scheme is based on a separate account of each constraint, the penalty function method not being used. The algorithm is substantiated and numerical experiments have confirmed its convergence and efficiency as compared with those ones which do not take into account the values of the first derivatives of the problem.
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