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## Local convergence of SQP methods for Mathematical Programs with Equilibrium Constraints (2002)

Citations: | 71 - 19 self |

### Citations

575 | SNOPT: An SQP algorithm for large-scale constrained optimization.
- Gill, Murray, et al.
- 2002
(Show Context)
Citation Context ...inconsistent. This is also observed during ourslter solves (we enter restoration at this point). Clearly, any NLP solver hoping to tackle MPECs will have to deal with this situation. The solver snopt =-=[12]-=- uses an elastic mode that relaxes the linearizations of the QP;slter [9] has a restoration phase. In Section 5 convergence of SQP methods without modications is analyzed. This analysis is closer in ... |

265 |
Mathematical Programs with Equilibrium Constraints.
- Luo, Pang, et al.
- 1996
(Show Context)
Citation Context ...e, University of Cambridge, Cambridge, CB2 1AG, UK, (fd.ralph,s.scholtesg@jims.cam.ac.uk). 1 2 Roger Fletcher, Sven Leyer, Danny Ralph, and Stefan Scholtes type arise frequently in applications; see =-=[7, 16, 17]-=- for references. (Problem (1.1) is also referred to as a mathematical program with complementarity constraints.) Clearly, an MPEC with a more general complementarity condition such as 0 G(z) ? H(z) ... |

243 | Nonlinear programming without a penalty function,”
- DENNIS, Fletcher, et al.
- 2002
(Show Context)
Citation Context ...ration at this point). Clearly, any NLP solver hoping to tackle MPECs will have to deal with this situation. The solver snopt [12] uses an elastic mode that relaxes the linearizations of the QP;slter =-=[9]-=- has a restoration phase. In Section 5 convergence of SQP methods without modications is analyzed. This analysis is closer in spirit to the results obtained usingslter. 3 Optimality Conditions for MP... |

188 | Engineering and Economic Applications of complementarity Problems,
- Ferris, Pang
- 1997
(Show Context)
Citation Context ...e, University of Cambridge, Cambridge, CB2 1AG, UK, (fd.ralph,s.scholtesg@jims.cam.ac.uk). 1 2 Roger Fletcher, Sven Leyer, Danny Ralph, and Stefan Scholtes type arise frequently in applications; see =-=[7, 16, 17]-=- for references. (Problem (1.1) is also referred to as a mathematical program with complementarity constraints.) Clearly, an MPEC with a more general complementarity condition such as 0 G(z) ? H(z) ... |

159 |
Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results
- Outrata, Kocvara, et al.
- 1998
(Show Context)
Citation Context ...e, University of Cambridge, Cambridge, CB2 1AG, UK, (fd.ralph,s.scholtesg@jims.cam.ac.uk). 1 2 Roger Fletcher, Sven Leyer, Danny Ralph, and Stefan Scholtes type arise frequently in applications; see =-=[7, 16, 17]-=- for references. (Problem (1.1) is also referred to as a mathematical program with complementarity constraints.) Clearly, an MPEC with a more general complementarity condition such as 0 G(z) ? H(z) ... |

148 |
Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity
- Scheel, Scholtes
(Show Context)
Citation Context ...1.3), obviously has no feasible point that satises the inequalities strictly. This fact implies that the Mangasarian-Fromovitz constraint qualication (MFCQ) is violated at every feasible point; see =-=[4, 19]-=-. There are other, MPEC-specic constraint qualications, such as the MPEC-LICQ explained below, which guarantee the existence of multipliers at local optima of (1.3) and are not overly stringent, see... |

82 |
Convergence properties of a regularization scheme for mathematical programs with complementarity constraints
- Scholtes
(Show Context)
Citation Context ... shows (k) = 2k1 + (k1)=2, the recurrence relation for . The explicit formula for follows easily. The iterates clearly converge linearly to the solution. Note that (P ) satises an MPEC-MFCQ =-=[20]-=- but violates an MPEC-LICQ (as can be seen easily by observing that four constraints are active at the solution). In addition, (P ) fails to satisfy strong complementarity. For strong complementarity,... |

49 |
The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions. Optimization 32
- Chen, Florian
- 1995
(Show Context)
Citation Context ...1.3), obviously has no feasible point that satises the inequalities strictly. This fact implies that the Mangasarian-Fromovitz constraint qualication (MFCQ) is violated at every feasible point; see =-=[4, 19]-=-. There are other, MPEC-specic constraint qualications, such as the MPEC-LICQ explained below, which guarantee the existence of multipliers at local optima of (1.3) and are not overly stringent, see... |

46 |
Perturbed Kuhn–Tucker points and rates of convergence for a class of nonlinear programming algorithms.Math
- ROBINSON
- 1974
(Show Context)
Citation Context ...d its corresponding multipliers (; 1; 2) are unique. Moreover, d is also the only strict local minimizer in a neighborhood of d = 0 of (QP k). Proof. The result for (QPR(z (k))) is due to Robinson =-=[18]-=- (see also Conn, Gould and Toint [6]), since the relaxed NLP satises [A1]{[A4]. The statement for (QP k) follows in two parts. First-order conditions are established in Lemma 5.5. Second-order condit... |

41 | Stabilized sequential quadratic programming
- Hager
- 1999
(Show Context)
Citation Context ... computes approximate KKT points for a sequence of active sets. The paper also complements the recently renewed interest in the convergence properties of SQP under weaker assumptions. See for example =-=[8, 13, 22]-=-. These studies suggest Local convergence of SQP methods for MPECs 3 modications to enable SQP solvers to handle NLP problems for which the constraint gradients are linearly dependent at the solution... |

40 | On solving mathematical programs with complementarity constraints as nonlinear programs
- Anitescu
- 2000
(Show Context)
Citation Context ...s 3 modications to enable SQP solvers to handle NLP problems for which the constraint gradients are linearly dependent at the solution and/or for which strict complementarity fails to hold. Anitescu =-=[1]-=- extends Wright's analysis [22] to NLPs with unbounded multiplier sets. The fact that (1.3) violates MFCQ implies that the multiplier set at stationary solutions will be unbounded. Anitescu's work the... |

40 | Solving mathematical programs with complementarity constraints as nonlinear programs.
- Fletcher, Leyffer
- 2004
(Show Context)
Citation Context ... projection method. Conn et al. [5] and Ferris and Pang [7] attribute certain failures of lancelot to the fact that the problem contains a complementarity constraint. In contrast, Fletcher and Leyer =-=[10]-=- recently reported encouraging numerical results on a large collection of MPECs [15]. They solved over 150 MPECs with an SQP solver and observed quadratic convergence for all but two problems. The two... |

38 | Modified SQP for degenerate problems
- Wright
- 1997
(Show Context)
Citation Context ... computes approximate KKT points for a sequence of active sets. The paper also complements the recently renewed interest in the convergence properties of SQP under weaker assumptions. See for example =-=[8, 13, 22]-=-. These studies suggest Local convergence of SQP methods for MPECs 3 modications to enable SQP solvers to handle NLP problems for which the constraint gradients are linearly dependent at the solution... |

34 |
Convex two-level optimization
- Bard
- 1988
(Show Context)
Citation Context ...ability of an NLP and the lack thereof has been advanced as a theoretical argument against the use of standard NLP solvers for MPECs. Numerical experience with (1.3) has also been disappointing. Bard =-=[2]-=- reports failure on 50{70% of some bilevel problems for a gradient projection method. Conn et al. [5] and Ferris and Pang [7] attribute certain failures of lancelot to the fact that the problem contai... |

33 |
How stringent is the linear independence assumption for mathematical programs with complementarity constraints
- Scholtes, Stöhr
(Show Context)
Citation Context ... There are other, MPEC-specic constraint qualications, such as the MPEC-LICQ explained below, which guarantee the existence of multipliers at local optima of (1.3) and are not overly stringent, see =-=[21]-=-. MFCQ, however, is a sucient condition for stability of an NLP and the lack thereof has been advanced as a theoretical argument against the use of standard NLP solvers for MPECs. Numerical experienc... |

29 | An implementable active-set algorithm for computing a B-stationary point of the mathematical program with linear complementarity constraints
- Fukushima, Tseng
(Show Context)
Citation Context ...have expressed renewed interest in the global convergence of algorithms for MPECs. Scholtes [20] analyzes a regularization scheme in which a sequence of parametric NLPs is solved. Fukushima and Tseng =-=[11]-=- analyze an algorithm that computes approximate KKT points for a sequence of active sets. The paper also complements the recently renewed interest in the convergence properties of SQP under weaker ass... |

23 | QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints
- Jiang, Ralph
- 1999
(Show Context)
Citation Context ...s to the AMPL model of the problem in MacMPEC, an AMPL collection of MPECs [15]. 2.1 Dependent Constraint Normals and Unbounded Multipliers In this section we use a small example from Jiang and Ralph =-=[14]-=- (see also jr*.mod) to illustrate the key idea of our approach. Consider the two MPECs( minimize z fi(z) subject to 0 z2 ? z2 z1 0 (2.1) with f1(z) = (z1 1) 2 + z22 and f2(z) = z 2 1 + (z2 1... |

22 | Modified Wilson method for nonlinear programs with nonunique multipliers
- Fischer
- 1999
(Show Context)
Citation Context ... computes approximate KKT points for a sequence of active sets. The paper also complements the recently renewed interest in the convergence properties of SQP under weaker assumptions. See for example =-=[8, 13, 22]-=-. These studies suggest Local convergence of SQP methods for MPECs 3 modications to enable SQP solvers to handle NLP problems for which the constraint gradients are linearly dependent at the solution... |

21 |
Numerical experiments with the LANCELOT package (Release A) for large-scale nonlinear optimization
- Toint
- 1996
(Show Context)
Citation Context ... standard NLP solvers for MPECs. Numerical experience with (1.3) has also been disappointing. Bard [2] reports failure on 50{70% of some bilevel problems for a gradient projection method. Conn et al. =-=[5]-=- and Ferris and Pang [7] attribute certain failures of lancelot to the fact that the problem contains a complementarity constraint. In contrast, Fletcher and Leyer [10] recently reported encouraging ... |

15 | Numerical experiments with the LANCELOT package (Release A) for large-scale nonlinear optimization - Conn, Gould, et al. - 1996 |

14 | SNOPT: An SQP algorithm for large scale constrained optimization - Gill, Murray, et al. - 2005 |

13 | On the solution of mathematical programming problems with equilibrium constraints - Andreani, Mart́ınez |

4 | Numerical experience with solving MPECs by nonlinear programming methods, Numerical Analysis Report NA/YYY - Fletcher, Leyffer - 2001 |

1 |
Local convergence ananlysis of newton-type methods for variational inequalities and nonlinear programming
- Bonnans
- 1994
(Show Context)
Citation Context ...re positivity of the biactive multipliers 1j and 2j, because Proposition 5.2, which underlies our convergence analysis there, does not require i 6= 0; 8i 2 E , and i > 0; 8i 2 A \ I, see =-=[3]-=-. Section 5.2 however requires the all the conditions of [A4]. Assumption [A5] is a reasonable assumption in practice, as most modern SQP solvers are based on active set QP solvers that guarantee this... |

1 | MacMPEC: AMPL collection of MPECs - Leyer - 2000 |

1 |
MacMPEC: AMPL collection of MPECs. Webpage
- Leyer
- 2000
(Show Context)
Citation Context ...es of lancelot to the fact that the problem contains a complementarity constraint. In contrast, Fletcher and Leyer [10] recently reported encouraging numerical results on a large collection of MPECs =-=[15]-=-. They solved over 150 MPECs with an SQP solver and observed quadratic convergence for all but two problems. The two problems that did not give quadratic convergence violate certain MPEC regularity co... |