## An implementable active-set algorithm for computing a Bstationary point of a mathematical program with linear complementarity constraints

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Venue: | SIAM J. Optim |

Citations: | 21 - 4 self |

### BibTeX

@ARTICLE{Fukushima_animplementable,

author = {Masao Fukushima and Paul Tseng},

title = {An implementable active-set algorithm for computing a Bstationary point of a mathematical program with linear complementarity constraints},

journal = {SIAM J. Optim},

year = {},

pages = {724--739}

}

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### Abstract

Abstract. In [3], an ɛ-active set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛ-feasible set, this algorithm generates iterates whose cluster points are B-stationary points of the problem. However, the proof has a gap and only shows that each cluster point is an M-stationary point. We discuss this gap and show that B-stationarity can be achieved if the algorithm is modified and an additional error bound condition holds. Key words. MPEC, B-stationary point, ɛ-active set, error bound AMS subject classifications. 65K05, 90C30, 90C33

### Citations

105 |
Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results
- Outrata, Kocvara, et al.
- 1998
(Show Context)
Citation Context ... function on ! n and G i ; H i ; g j ; h l are real-valued affine functions on ! n . This problem has been of much interest and many algorithms have been proposed for its solution, as is evidenced by =-=[1, 2, 5, 7, 9]-=- and the extensive references therein. However, these algorithms in general are only guaranteed to compute either a B-stationary point under the nondegeneracy (strict complementarity) assumption that ... |

97 |
Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity
- Scheel, Scholtes
(Show Context)
Citation Context ...empts to identify a favorable class of 2 MPECs have been focused on constraint qualifications pertaining to MPEC which enable us to characterize a B-stationary point in a simple and convenient manner =-=[10, 13]-=-. For each z 2 F ffl , define the index sets: A ffl (z) := fi : G i (z)sfflg; B ffl (z) := fi : H i (z)sfflg; I ffl (z) := fj : g j (z)s\Gammafflg: By the definition of P, we always have A ffl (z) [ B... |

90 |
On approximate solutions of systems of linear inequalities
- Hoffman
- 1952
(Show Context)
Citation Context ... [ K 0 , ~ z k satisfies all constraints defining F ffl k+1 [ ~ A k ; ~ B k ], except for possibly G i (z)sffl k+1 8 i 2 ~ A k ; H i (z)sffl k+1 8 i 2 ~ B k : 8 Then, by a well-known lemma of Hoffman =-=[4]-=-, there exists a constants? 0 (depending on rG i ; rH i ; rg j ; rh l only) such that kz k+1 \Gamma ~ z k ks0 @ X i2 ~ A k fi fi fi[G i (~z k ) \Gamma ffl k+1 ] + fi fi fi + X i2 ~ B k fi fi fi[H i (~... |

59 |
Convergence properties of a regularization scheme for mathematical programs with complementarity constraints
- Scholtes
(Show Context)
Citation Context ...c weak nondegeneracy" assumption, convergence of second-order stationary point of the smoothed problem to a B-stationary point of MPEC as the smoothing parameter tends to zero. Subsequently, Scho=-=ltes [11]-=- showed an analogous result for a method based on another smoothed approximation of MPEC. However, the algorithms considered in [3, 11] are conceptual, and global convergence to a B-stationary point h... |

52 | A smoothing method for mathematical programs with equilibrium constraints
- Facchinei, Jiang, et al.
- 1999
(Show Context)
Citation Context ... function on ! n and G i ; H i ; g j ; h l are real-valued affine functions on ! n . This problem has been of much interest and many algorithms have been proposed for its solution, as is evidenced by =-=[1, 2, 5, 7, 9]-=- and the extensive references therein. However, these algorithms in general are only guaranteed to compute either a B-stationary point under the nondegeneracy (strict complementarity) assumption that ... |

44 | Error Bound and Convergence Analysis of Matrix Splitting Algorithms for the Affine Variational Inequality Problem
- Luo, Tseng
- 1992
(Show Context)
Citation Context ...k 2 K, are distinct. On the other hand,sk = 0 implies ffi k = 0 for all k , so (5) yields thatsz k is an (exact) stationary point of the subproblem (3). Since (3) is a quadratic program, a lemma from =-=[8]-=- shows that the set of values f f(z) : z is a stationary point of (3) g is finite. Since ffl k = 0 for all k and the number of index set pairs (A; B) 2 P is finite, the number of distinct quadratic pr... |

40 | On the accurate identification of active constraints
- Facchinei, Fischer, et al.
- 1998
(Show Context)
Citation Context ...y assert that Ā ∩ ¯ B ⊆ A0(¯z) ∩ B0(¯z). This is the gap. 2. A modified ɛ-active set algorithm. We now describe a way, based on an active set identification approach of Facchinei, Fischer, and Kanzow =-=[1]-=-, to modify the ɛ-active set algorithm so that (3) holds under an additional error bound condition. To simplify the notation, we will consider only the complementarity constraints, i.e., we assume p =... |

38 |
A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints
- Fukushima, Luo, et al.
- 1998
(Show Context)
Citation Context ... function on ! n and G i ; H i ; g j ; h l are real-valued affine functions on ! n . This problem has been of much interest and many algorithms have been proposed for its solution, as is evidenced by =-=[1, 2, 5, 7, 9]-=- and the extensive references therein. However, these algorithms in general are only guaranteed to compute either a B-stationary point under the nondegeneracy (strict complementarity) assumption that ... |

36 |
Exact penalization of mathematical programs with equilibrium constraints
- Scholtes, Stohr
- 1999
(Show Context)
Citation Context ...the nondegeneracy (strict complementarity) assumption that is somewhat restrictive in practice, or a C-stationary point for the problem, rather than the desired B-stationary point. Scholtes and Stohr =-=[12]-=- showed that a trust region method converges to a B-stationary point provided the trust region radii do not tend to zero. However, it is not clear whether we can expect the latter condition to hold in... |

34 | Smooth SQP methods for mathematical programs with nonlinear complementarity constraints
- Jiang, Ralph
- 2000
(Show Context)
Citation Context |

27 |
Convergence of a smoothing continuation method for mathematical programs with complementarity constraints
- Fukushima, Pang
- 1999
(Show Context)
Citation Context ...-stationary point provided the trust region radii do not tend to zero. However, it is not clear whether we can expect the latter condition to hold in the degenerate case. Recently, Fukushima and Pang =-=[3]-=- considered a continuation method based on a smoothed 1 This research is supported by Scientific Research Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan, and Nationa... |

21 | Optimality conditions for a class of mathematical programs with equilibrium constraints - Outrata - 1999 |

20 | QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints
- Jiang, Ralph
- 1999
(Show Context)
Citation Context ...from the Matlab Optimization Toolbox. We stop at Step 3 of iteration k when v k is0; w k is0 for all i 2 A k " B k . In our testing, we use the Matlab program QPECgen (Version 1.1) of Jiang and R=-=alph [6]-=- to generate test problems. This program generates random instances of mathematical programs with quadratic objective function and affine variational inequality constraints. We set the QPECgen paramet... |

15 | Complementarity constraint qualifications and simplified B-stationary conditions for mathematical programs with equilibrium constraints
- FUKUSHIMA, PANG
- 1999
(Show Context)
Citation Context ...empts to identify a favorable class of 2 MPECs have been focused on constraint qualifications pertaining to MPEC which enable us to characterize a B-stationary point in a simple and convenient manner =-=[10, 13]-=-. For each z 2 F ffl , define the index sets: A ffl (z) := fi : G i (z)sfflg; B ffl (z) := fi : H i (z)sfflg; I ffl (z) := fj : g j (z)s\Gammafflg: By the definition of P, we always have A ffl (z) [ B... |

10 |
On approximate solutions of systems of linear inequalities
- Homan
- 1952
(Show Context)
Citation Context ...or each k # K # K # ,sz k satisfies all constraints defining F # k+1 [ A k , B k ], except for possibly G i (z) # # k+1 # i #sA k , H i (z) # # k+1 # i # B k . 8 Then, by a well-known lemma of Ho#man =-=[4]-=-, there exists a constant # > 0 (depending on #G i , #H i , #g j , #h l only) such that #z k+1 -sz k # # # 0 @ X i#sA ks[G i (z k ) - # k+1 ] + + X i#sB ks[H i (z k ) - # k+1 ] + 1 A = # 0 @ X i#sA ks... |