## Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints (2005)

Venue: | Industrial Engineering & Management Sciences, Northwestern University |

Citations: | 15 - 1 self |

### BibTeX

@TECHREPORT{Liu05generalizedstationary,

author = {Xinwei Liu and Jie Sun},

title = {Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints},

institution = {Industrial Engineering & Management Sciences, Northwestern University},

year = {2005}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primal-dual interior-point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or the linear independence constraint qualification for MPEC (MPEC-LICQ). Under certain general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interior-point algorithm failed to find a stationary point. Key words: Global convergence, interior-point methods, mathematical programming with equilibrium constraints, stationary point

### Citations

405 |
AMPL: A Modeling Language for
- Fourer, Gay, et al.
- 2002
(Show Context)
Citation Context ... iterate goes beyond the domain of some function used, so the solution process is adjourned. The details on using MATLAB with AMPL can be found in Gay’s preprint [19] and in the book of Fourer et al. =-=[14]-=-. The computational experiments have been done on a Hewlett Packard C3600 workstation with the UNIX system. Many test problems in the former subsection are included in this test suite, it is still wor... |

195 |
mathematical programs with equilibrium constraints
- Luo, JS, et al.
- 1996
(Show Context)
Citation Context ...s of very difficult optimization problems [4, 7, 8]. There have been many papers dealing with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example =-=[16, 22, 24, 32, 34, 36, 40]-=-, while some other papers proposed algorithms for MPEC, see [10, 15, 17, 18, 25, 32, 33, 35, 37, 39]. Upon the success of interior-point methods for linear and nonlinear programming (NLP), the interio... |

111 |
Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity
- Scheel, Scholtes
(Show Context)
Citation Context ...ffer, Ralph, and Scholtes [13]) and, on the other hand, it brings new challenges to the NLP algorithm to be used. In particular, since MPECs violate the Mangasarian-Fromovitz constraint qualification =-=[7, 38]-=-, the usual conditions for global convergence of an NLP algorithm may not be met and the linearized complementarity constraints may be inconsistent when the iterates close to the solutions (see [28] f... |

97 | Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods
- Shanno, Vanderbei
(Show Context)
Citation Context ...ollapse at some iterates. As a result, PIPA may fail to find a stationary point for a simple MPCC. An interesting idea of designing algorithms for MPEC, discussed in a series of recent studies, e.g., =-=[5, 9, 10, 12, 27]-=-, is to reformulate the MPEC as an NLP problem or a sequence of NLP problems and to solve the reformulated problem(s) by various algorithms of NLP. On one hand, this approach allows us to take advanta... |

71 |
Convergence properties of a regularization scheme for mathematical programs with complementarity constraints
- Scholtes
(Show Context)
Citation Context ... with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example [16, 22, 24, 32, 34, 36, 40], while some other papers proposed algorithms for MPEC, see =-=[10, 15, 17, 18, 25, 32, 33, 35, 37, 39]-=-. Upon the success of interior-point methods for linear and nonlinear programming (NLP), the interiorpoint approach has been extended to solve MPEC as well. The penalty interior-point algorithm (PIPA)... |

65 |
A primal-dual interior method for nonconvex nonlinear programming
- Gay, Overton, et al.
- 1997
(Show Context)
Citation Context ...ll test problems. This way of updating the barrier parameter has been used in a number of interior-point algorithms for NLP such as Byrd, Hribar and Nocedal [6]. Some other strategies are reported in =-=[20, 21]-=-. Our preliminary experiment appears to show that there is no obvious impact on the number of iterations if we change from the fixed factor to a variable factor. One of the reasons may be that the sol... |

64 | A smoothing method for mathematical programs with equilibrium constrains
- Facchinei, Jiang, et al.
- 1999
(Show Context)
Citation Context ... with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example [16, 22, 24, 32, 34, 36, 40], while some other papers proposed algorithms for MPEC, see =-=[10, 15, 17, 18, 25, 32, 33, 35, 37, 39]-=-. Upon the success of interior-point methods for linear and nonlinear programming (NLP), the interiorpoint approach has been extended to solve MPEC as well. The penalty interior-point algorithm (PIPA)... |

59 | Local convergence of SQP methods for mathematical programs with equilibrium constraints
- Fletcher, Leyffer, et al.
- 2002
(Show Context)
Citation Context ...problem(s) by various algorithms of NLP. On one hand, this approach allows us to take advantage of certain NLP algorithms to obtain rapid local convergence (see Fletcher, Leyffer, Ralph, and Scholtes =-=[13]-=-) and, on the other hand, it brings new challenges to the NLP algorithm to be used. In particular, since MPECs violate the Mangasarian-Fromovitz constraint qualification [7, 38], the usual conditions ... |

57 |
An interior point algorithm for large-scale nonlinear programming
- Byrd, Hribar, et al.
- 1999
(Show Context)
Citation Context ...e the number of outer iterations is 7 for all test problems. This way of updating the barrier parameter has been used in a number of interior-point algorithms for NLP such as Byrd, Hribar and Nocedal =-=[6]-=-. Some other strategies are reported in [20, 21]. Our preliminary experiment appears to show that there is no obvious impact on the number of iterations if we change from the fixed factor to a variabl... |

57 | Bilevel and multilevel programming: A bibliography review
- Vicente, Calamai
- 1994
(Show Context)
Citation Context ...ric nonlinear complementarity problem, in which case MPEC is specifically called the mathematical program with complementarity constraints (MPCC). MPEC includes the bilevel programming problem (e.g., =-=[11, 41]-=-) as its special case and has extensive applications in practical areas such as traffic control, engineering design, and economic modeling, see [2, 23, 32, 33]. Since there are variational inequalitie... |

43 |
A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints
- Fukushima, Luo, et al.
- 1998
(Show Context)
Citation Context ... with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example [16, 22, 24, 32, 34, 36, 40], while some other papers proposed algorithms for MPEC, see =-=[10, 15, 17, 18, 25, 32, 33, 35, 37, 39]-=-. Upon the success of interior-point methods for linear and nonlinear programming (NLP), the interiorpoint approach has been extended to solve MPEC as well. The penalty interior-point algorithm (PIPA)... |

41 |
Exact penalization of mathematical programs with equilibrium constraints
- Scholtes, Stöhr
- 1999
(Show Context)
Citation Context |

39 |
The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions
- Chen, Florian
- 1995
(Show Context)
Citation Context ...x, nonsmooth, disconnected, and non-closed even if all involved functions have very good analytical properties (see [32]). As such, MPEC is known to be a class of very difficult optimization problems =-=[4, 7, 8]-=-. There have been many papers dealing with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example [16, 22, 24, 32, 34, 36, 40], while some other pape... |

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

33 |
Convergence of a smoothing continuation method for mathematical problem with complementarity constraints
- Fukushima, Pang
- 1999
(Show Context)
Citation Context |

29 | Hooking your solver to AMPL
- Gay
- 1997
(Show Context)
Citation Context ... “out of domain”, meaning that certain iterate goes beyond the domain of some function used, so the solution process is adjourned. The details on using MATLAB with AMPL can be found in Gay’s preprint =-=[19]-=- and in the book of Fourer et al. [14]. The computational experiments have been done on a Hewlett Packard C3600 workstation with the UNIX system. Many test problems in the former subsection are includ... |

29 |
A numerical approach to optimization problems with variational inequality constraints
- Outrata, Zowe
- 1995
(Show Context)
Citation Context |

28 |
Convex two-level optimization
- Bard
- 1988
(Show Context)
Citation Context ... a Pentium-III 450MHz processor and WINDOWS98 operating system. We first applied our algorithms to the set of test problems listed in the Appendix of [10]. Some of the test problems were also used in =-=[1, 3, 34, 35, 39]-=- to test various algorithms developed for MPEC. 20The test problem 7 has a nondifferentiable term w = max{0, x1 + x2 + y1 − 2y2 − 40} in the objective function, we reformulate it as a smooth problem ... |

23 | An implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints
- Fukushima, Tseng
(Show Context)
Citation Context |

23 |
Network design problem with congestion effects: a case of bilevel programming
- Marcotte
- 1986
(Show Context)
Citation Context ...includes the bilevel programming problem (e.g., [11, 41]) as its special case and has extensive applications in practical areas such as traffic control, engineering design, and economic modeling, see =-=[2, 23, 32, 33]-=-. Since there are variational inequalities in the constraints of the problem, the feasible region may be nonconvex, nonsmooth, disconnected, and non-closed even if all involved functions have very goo... |

21 | On bilevel programming, Part I: general nonlinear cases - Falk, Liu - 1995 |

21 |
On optimality problems with variational inequality constraints
- Outrata
(Show Context)
Citation Context ...s of very difficult optimization problems [4, 7, 8]. There have been many papers dealing with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example =-=[16, 22, 24, 32, 34, 36, 40]-=-, while some other papers proposed algorithms for MPEC, see [10, 15, 17, 18, 25, 32, 33, 35, 37, 39]. Upon the success of interior-point methods for linear and nonlinear programming (NLP), the interio... |

19 | Some feasibility issues in mathematical programs with equilibrium constraints
- Fukushima, Pang
- 1998
(Show Context)
Citation Context ...s of very difficult optimization problems [4, 7, 8]. There have been many papers dealing with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example =-=[16, 22, 24, 32, 34, 36, 40]-=-, while some other papers proposed algorithms for MPEC, see [10, 15, 17, 18, 25, 32, 33, 35, 37, 39]. Upon the success of interior-point methods for linear and nonlinear programming (NLP), the interio... |

17 |
On the existence of optimal solutions to mathematical program with equilibrium constraints
- Harker, Pang
- 1988
(Show Context)
Citation Context |

17 | Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints
- Pang, Fukushima
- 1999
(Show Context)
Citation Context |

14 | Mathematical programs with equilibrium constraints: Automatic reformulation and solution via constrained optimization. Numerical Analysis Group Research Report NA-02/11
- Ferris, Dirkse, et al.
- 2002
(Show Context)
Citation Context ...ollapse at some iterates. As a result, PIPA may fail to find a stationary point for a simple MPCC. An interesting idea of designing algorithms for MPEC, discussed in a series of recent studies, e.g., =-=[5, 9, 10, 12, 27]-=-, is to reformulate the MPEC as an NLP problem or a sequence of NLP problems and to solve the reformulated problem(s) by various algorithms of NLP. On one hand, this approach allows us to take advanta... |

14 | A robust primal-dual interior point algorithm for nonlinear programs
- Liu, Sun
(Show Context)
Citation Context ...gested to overcome difficulties in global convergence. In this paper we present an interior-point method for MPEC. The method, together with its convergence theory, is an extension of a robust method =-=[29, 31]-=- developed by the authors for NLP. The original motivation of that research was to develop an algorithm that can handle some “bad” cases of NLP. The idea and analysis used in that research turn out to... |

13 | Superlinear convergence of primal-dual interior point algorithms for nonlinear programming
- Gould, Orban, et al.
- 2001
(Show Context)
Citation Context ...ll test problems. This way of updating the barrier parameter has been used in a number of interior-point algorithms for NLP such as Byrd, Hribar and Nocedal [6]. Some other strategies are reported in =-=[20, 21]-=-. Our preliminary experiment appears to show that there is no obvious impact on the number of iterations if we change from the fixed factor to a variable factor. One of the reasons may be that the sol... |

12 |
Hierarchical decentralized systems and its new solution by a barrier method
- Aiyoshi, Shimizu
- 1981
(Show Context)
Citation Context ...Otherwise, we replace ρk by a larger ρk+1 (for example ρk+1 ≥ 2ρk) such that (4.20) holds; 11Step 4 Compute ˆαk ∈ (0, 1] such that z k + ˆαkd k z ≥ ξz k , and select firstly σ ∈ (0, 1] and then γk ∈ =-=[0, 1]-=- as large as possible such that φ(s k + σ ˆαkd k s, z k + σ ˆαkd k z; ρk+1) − φ(s k , z k ; ρk+1) ≤ σ0σ ˆαkπk(d k ; ρk+1), (4.21) β1µe ≤ (Uk + γkD k u) max{z k + σ ˆαkd k z, −Gθ(s k + σ ˆαkd k s)} ≤ β... |

11 | Complementarity Constraints as Nonlinear Equations: Theory and Numerical Experiences
- Leyffer
- 2003
(Show Context)
Citation Context ...7, 38], the usual conditions for global convergence of an NLP algorithm may not be met and the linearized complementarity constraints may be inconsistent when the iterates close to the solutions (see =-=[28]-=- for a detailed analysis). 1Therefore, a successful NLP algorithm should be able to handle the irregularity of the reformulated problem. An example of such algorithms is due to Fletcher and Leyffer [... |

9 |
Computational difficulties of bilevel linear programming
- Ben-Ayed, Blair
- 1990
(Show Context)
Citation Context ...x, nonsmooth, disconnected, and non-closed even if all involved functions have very good analytical properties (see [32]). As such, MPEC is known to be a class of very difficult optimization problems =-=[4, 7, 8]-=-. There have been many papers dealing with MPEC in recent years. Some of them considered the existence and stationarity of its solution, for example [16, 22, 24, 32, 34, 36, 40], while some other pape... |

6 | A robust algorithm for optimization with general equality and inequality constraints
- Liu, Yuan
(Show Context)
Citation Context ...gested to overcome difficulties in global convergence. In this paper we present an interior-point method for MPEC. The method, together with its convergence theory, is an extension of a robust method =-=[29, 31]-=- developed by the authors for NLP. The original motivation of that research was to develop an algorithm that can handle some “bad” cases of NLP. The idea and analysis used in that research turn out to... |

5 | The penalty interior point method fails to converge for mathematical programs with equilibrium constraints. University of Dundee Report NA208
- Leyffer
- 2002
(Show Context)
Citation Context ...erior-point method for MPEC. Its global convergence requires the linear independence constraint qualification for MPEC (MPECLICQ) and strict complementarity. However, it was found recently by Leyffer =-=[26]-=- that some conditions required by PIPA for convergence may collapse at some iterates. As a result, PIPA may fail to find a stationary point for a simple MPCC. An interesting idea of designing algorith... |

4 |
Numerical experience with solving MPECs by nonlinear programming methods,” Numerical Analysis
- Fletcher, Leyffer
(Show Context)
Citation Context ...ollapse at some iterates. As a result, PIPA may fail to find a stationary point for a simple MPCC. An interesting idea of designing algorithms for MPEC, discussed in a series of recent studies, e.g., =-=[5, 9, 10, 12, 27]-=-, is to reformulate the MPEC as an NLP problem or a sequence of NLP problems and to solve the reformulated problem(s) by various algorithms of NLP. On one hand, this approach allows us to take advanta... |

3 | Global convergence analysis of line search interior point methods for nonlinear programming without regularity assumptions, accepted for publication in - Liu, Sun |