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**On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints.**
*(English)*
Zbl 1097.90050

Summary: We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using algorithms and procedures of smooth nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sufficient conditions for their Lagrange multiplier set to be nonempty. MPCCs that have nonempty Lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter. In this context, the elastic mode transforms MPCC into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem, which in turn makes it approachable by sequential quadratic programming (SQP) algorithms. One such algorithm is shown to achieve local linear convergence once the problem is relaxed. Under stronger conditions, we also prove superlinear convergence to the solution of an MPCC using an adaptive elastic mode approach for an SQP algorithm recently analyzed in an MPCC context in [R. Fletcher, S. Leyffer, D. Ralph and S. Scholtes, Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints, SIAM J. Optim. 17, No. 1, 2589–286 (2006; Zbl 1112.90098)]. Our assumptions are more general since we do not use a critical assumption from that reference. In addition, we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen.

### MSC:

90C30 | Nonlinear programming |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C55 | Methods of successive quadratic programming type |

49M37 | Numerical methods based on nonlinear programming |

65K10 | Numerical optimization and variational techniques |