Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties.

*(English)*Zbl 1119.90050Authors’ summary: The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first- or second-order point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, M-stationarity, or C-stationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic-mode formulation is solved inexactly.

Reviewer: Leszek S. Zaremba (Warszawa)

##### MSC:

90C30 | Nonlinear programming |

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

65K05 | Numerical mathematical programming methods |

##### Keywords:

Nonlinear programming; Equilibrium constraints; Complementarity constraints; Elastic-mode formulation; Strong stationarity; C-stationarity; M-stationarity
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\textit{M. Anitescu} et al., Math. Program. 110, No. 2 (A), 337--371 (2007; Zbl 1119.90050)

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##### References:

[1] | Anitescu M. (2005) On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints. SIAM J. Optim. 15, 1203–1236 · Zbl 1097.90050 |

[2] | Anitescu M. (2005) Global convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints. SIAM J. Optim. 16, 120–145 · Zbl 1099.65050 |

[3] | DeMiguel A., Friedlander M., Nogales F.J., Scholtes S.: An interior-point method for MPECs based on strictly feasible relaxations, Preprint ANL/MCS-P1150-0404, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, USA, Argonne, IL, USA (2004) |

[4] | Fletcher R., Leyffer S. (2002) Nonlinear programming without a penalty function. Math. Program. Ser. A 91, 239–269 · Zbl 1049.90088 |

[5] | Fletcher R., Leyffer S.: Numerical experience with solving MPECs as NLPs, Report NA/210, University of Dundee (2002) |

[6] | Fukushima M., Pang J.-S.: Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: Théra, M., Tichatschke R. (eds.) Ill-posed Variational Problems and Regularization Techniques, vol. 477 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin Heidelberg New York, pp. 99–110 (1999) · Zbl 0944.65070 |

[7] | Fukushima M., Tseng P. (2002) An implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints. SIAM J. Optim. 12, 724–739 · Zbl 1005.65064 |

[8] | Gay D.M.: Hooking your solver to AMPL, technical report, Bell Laboratories. Murray Hill, NJ, 1993. Revised 1994, 1997 |

[9] | Hu X.M., Ralph D. (2004) Convergence of a penalty method for mathematical programming with complementarity constraints. J. Optim. Theory Appl. 123, 365–390 |

[10] | Jongen H.T., Jonker P., Twilt F. (1986) Critical sets in parametric optimization. Math. Program. 34, 333–353 · Zbl 0599.90114 |

[11] | Jongen H.T., Jonker P., Twilt F. (1986) One-parameter families of optimization problems: Equality constraints. J Optim. Theory Appl. 48, 141–161 · Zbl 0556.90086 |

[12] | Kocvara M., Outrata J. (2004) Optimization problems with equilibrium constraints and their numerical solution. Math. Program. 101, 119–149 · Zbl 1076.90058 |

[13] | Leyffer S.: MacMPEC AMPL collection of mathematical programs with equilibrium constraints |

[14] | Lin G.H., Fukushima M. (2003) New relaxation method for mathematical programs with complementarity constraints. J. Optim. Theory Appl. 118, 81–116 · Zbl 1033.90086 |

[15] | Liu X., Sun J. (2004) Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints. Math. Program. 101, 231–261 · Zbl 1076.90059 |

[16] | Luo Z.-Q., Pang J.-S., Ralph D.: Piecewise sequential quadratic programming for mathematical programs with nonlinear complementarity constraints. In: Migdalas A., Pardalos P., Värbrand P. (eds.) Multilevel Optimization: Algorithms, Complexity and Applications. Kluwer, Dordrecht, pp. 209–229 (1998) · Zbl 0897.90184 |

[17] | Outrata J.(1999) Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24, 627–644 · Zbl 1039.90088 |

[18] | Outrata J., Kocvara M., Zowe J. (1998) Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications, and Numerical Results. Kluwer, Dordrecht · Zbl 0947.90093 |

[19] | Overton M.L.: Numerical Computing with IEEE Floating-Point Arithmetic. SIAM (2001) · Zbl 0981.68057 |

[20] | Ralph D., Wright S.J. (2004) Some properties of regularization and penalization schemes for MPECs. Optim. Methods Softw. 19, 527–556 · Zbl 1097.90054 |

[21] | Robinson S.M. (1982) Generalized equations and their solutions. Part II: Applications to nonlinear programming. Math. Program. Study 19, 200–221 · Zbl 0495.90077 |

[22] | Scheel H., Scholtes S. (2000) Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 · Zbl 1073.90557 |

[23] | Scholtes S. (2001) Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 · Zbl 1010.90086 |

[24] | Scholtes S., Stöhr M. (2001) How stringent is the linear independence assumption for mathematical programs with complementarity constraints? Math. Oper. Res. 26, 851–863 · Zbl 1082.90580 |

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