## Complementarity constraints as nonlinear equations: Theory and numerical experience (2003)

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Venue: | Preprint ANL/MCS-P1054-0603, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne |

Citations: | 11 - 4 self |

### BibTeX

@INPROCEEDINGS{Leyffer03complementarityconstraints,

author = {Sven Leyffer},

title = {Complementarity constraints as nonlinear equations: Theory and numerical experience},

booktitle = {Preprint ANL/MCS-P1054-0603, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne},

year = {2003},

pages = {169--208},

publisher = {Springer}

}

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### Abstract

Recently, it has been shown that mathematical programs with complementarity constraints (MPCCs) can be solved efficiently and reliably as nonlinear programs. This paper examines various nonlinear formulations of the complementarity constraints. Several nonlinear complementarity functions are considered for use in MPCC. Unlike standard smoothing techniques, however, the reformulations do not require the control of a smoothing parameter. Thus they have the advantage that the smoothing is exact in the sense that Karush-Kuhn-Tucker points of the reformulation correspond to strongly stationary points of the MPCC. A new exact smoothing of the well-known min function is also introduced and shown to possess desirable theoretical properties. It is shown how the new formulations can be integrated into a sequential quadratic programming solver, and their practical performance is compared on a range of test problems.

### Citations

410 |
AMPL: A Modeling Language for
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- 1992
(Show Context)
Citation Context ...filter. The filter accepts a trial point whenever the objective or the constraint violation is improved compared with all previous iterates [14, 15, 18]. 6.1 Preliminaries The solver includes an AMPL =-=[19]-=- interface that introduces slacks to formulate general complementarity constraints in the form (1.1) and handles the reformulation to the NLP (1.3) automatically. The interface also computes the deriv... |

374 | SNOPT: An SQP algorithm for large-scale constrained optimization
- Gill, Murray, et al.
- 2002
(Show Context)
Citation Context ...j ∈ D∗. [A5] The QP solver always chooses a linearly independent basis. We note that [A0] is readily implemented and that assumption [A5] holds for the QP solvers used within snopt due to Gill et al. =-=[GMS02]-=- and filter [FL02]. The most restrictive assumptions are [A2] and [A3] because they exclude B-stationary points that are not strongly stationary. This fact is not surprising because it is well known t... |

198 |
Mathematical Programs with Equilibrium Constraints
- Luo, Pang, et al.
- 1997
(Show Context)
Citation Context ...blems, giving rise to mathematical programs with complementarity constraints (MPCCs). Problems of this type arise in many engineering and economic applications; see the survey [11] and the monographs =-=[24, 26]-=-. The growing collections of test problems [22, 7] indicate that this an important area. MPCCs can be ∗ Preprint ANL/MCS-P1054-0603 1s2 Sven Leyffer expressed in general as minimize f(x) (1.1a) subjec... |

179 | Nonlinear programming without a penalty function
- Fletcher, Leyer
(Show Context)
Citation Context ...e (QP k ) for a step d Set x k+1 = x k + d, and k = k + 1 Algorithm 1: Local SQP Algorithm for MPCCs In practice, we also include a globalization scheme to stabilize SQP. In our case, we use a filter =-=[15]-=- and a trust region to ensure convergence to stationary points [18]. The convergence theory of filter methods allows for three possible outcomes [18, Theorem 1]: (A) The algorithm terminates at a poin... |

150 | Engineering and Economic Applications of Complementarity Problems
- Ferris, Pang
- 1997
(Show Context)
Citation Context ...ints in optimization problems, giving rise to mathematical programs with complementarity constraints (MPCCs). Problems of this type arise in many engineering and economic applications; see the survey =-=[11]-=- and the monographs [24, 26]. The growing collections of test problems [22, 7] indicate that this an important area. MPCCs can be ∗ Preprint ANL/MCS-P1054-0603 1s2 Sven Leyffer expressed in general as... |

122 |
Nonsmooth Approach to Optimization Problems with Equilibrium Constraints
- Outrata, Koc̆vara, et al.
- 1998
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Citation Context ...blems, giving rise to mathematical programs with complementarity constraints (MPCCs). Problems of this type arise in many engineering and economic applications; see the survey [11] and the monographs =-=[24, 26]-=-. The growing collections of test problems [22, 7] indicate that this an important area. MPCCs can be ∗ Preprint ANL/MCS-P1054-0603 1s2 Sven Leyffer expressed in general as minimize f(x) (1.1a) subjec... |

118 | Practical Methods of Optimization, 2nd edition - Fletcher - 2000 |

113 |
Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity
- Scheel, Scholtes
(Show Context)
Citation Context ...ransforms the MPCC into an equivalent nonlinear program (NLP) and is appealing because it appears to allow standard large-scale NLP solvers to be used to solve (1.1). Unfortunately, it has been shown =-=[5, 29]-=- that (1.2) violates the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. This failure of MFCQ has a number of unpleasant consequences: The multiplier set is unbounded, the... |

95 | Interior-point methods for nonconvex nonlinear programming: filter methods and merit functions
- Benson, Shanno, et al.
(Show Context)
Citation Context ...iently large penalty parameter can be found, and standard SQP methods converge. The convergence properties of interior point methods (IPMs) have also received renewed attention. Numerical experiments =-=[3, 28]-=- have shown that IPMs with minor modifications can be applied successfully to solve MPCCs. This practical success has encouraged theoretical studies of the convergence properties of IPMs for MPCCs. Ra... |

64 | A smoothing method for mathematical programs with equilibrium constrains
- Facchinei, Jiang, et al.
- 1999
(Show Context)
Citation Context ...mooth at the origin. We will show that this nonsmoothness does not affect the local convergence properties of the SQP method. The use of NCP functions for the solution of MPCCs has been considered in =-=[8, 10]-=-, where a sequence of smoothed NCP reformulation is solved. Our contribution is to show that this smoothing is not required. Thus we avoid the need to control the smoothing parameter that may be probl... |

60 | Local convergence of SQP methods for mathematical programs with equilibrium constraints
- Fletcher, Leyffer, et al.
(Show Context)
Citation Context ... a large class of MPCCs, written as NLPs, reliably and efficiently [16]. This numerical success has motivated a closer investigation of the (local) convergence properties of SQP methods for MPCCs. In =-=[17]-=-, it is shown that an SQP method converges locally to strongly stationary points. Anitescu [1] establishes that an SQP method with elastic mode converges locally for MPCCs with (1.2). The key idea is ... |

52 |
Smooth approximation to nonlinear complementarity problems
- Chen, Harker
- 1997
(Show Context)
Citation Context ...cher-Burmeister function [12] is given by φ FB(a,b) = a + b − √ a 2 + b 2 . (2.5)s4 Sven Leyffer It is nondifferentiable at the origin, and its Hessian is unbounded at the origin. 3. The min-function =-=[6]-=- is the nonsmooth function φmin(a,b) = min(a,b). (2.6) It can be written equivalently in terms of the natural residual function [6]: φNR(a,b) = 1 � a + b − 2 � (a − b) 2 � . (2.7) This function is aga... |

49 | A penalized Fischer-Burmeister NCPfunction
- Chen, Chen, et al.
(Show Context)
Citation Context ... of the natural residual function [6]: φNR(a,b) = 1 � a + b − 2 � (a − b) 2 � . (2.7) This function is again nondifferentiable at the origin and along the line a = b. 4. The Chen-Chen-Kanzow function =-=[4]-=- is a convex combination of the Fischer-Burmeister function and the bilinear function. For a fixed parameter λ ∈ (0, 1), it is defined as φ CCK(a,b) = λφ FB(a,b) + (1 − λ)a+b+, where a+ = max(0,a). No... |

42 |
M,(1995) The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions
- Chen, Florian
(Show Context)
Citation Context ...ransforms the MPCC into an equivalent nonlinear program (NLP) and is appealing because it appears to allow standard large-scale NLP solvers to be used to solve (1.1). Unfortunately, it has been shown =-=[5, 29]-=- that (1.2) violates the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. This failure of MFCQ has a number of unpleasant consequences: The multiplier set is unbounded, the... |

34 | On solving mathematical programs with complementarity constraints as nonlinear programs
- Anitescu
- 2001
(Show Context)
Citation Context ... has motivated a closer investigation of the (local) convergence properties of SQP methods for MPCCs. In [17], it is shown that an SQP method converges locally to strongly stationary points. Anitescu =-=[1]-=- establishes that an SQP method with elastic mode converges locally for MPCCs with (1.2). The key idea is to penalize X1x2 ≤ 0 and consider the resulting NLP, which satisfies MFCQ. Near a strongly sta... |

29 |
Solving mathematical program with complementarity constraints as nonlinear programs
- Fletcher, Leyffer
- 2004
(Show Context)
Citation Context ...prognosis about the use of (2) may have been premature. We have used standard sequential quadratic programming (SQP) solvers to solve a large class of MPCCs, written as NLPs, reliably and efficiently =-=[FL04]-=-. This numerical success has motivated a closer investigation of the (local) convergence properties of SQP methods for MPCCs. Fletcher et al. [FLRS02] show that an SQP method converges locally to stro... |

28 |
Convex two-level optimization
- Bard
(Show Context)
Citation Context ...constraint normals are linearly dependent, and linearizations of the NLP can be inconsistent arbitrarily close to a solution. In addition, early numerical experience with (1.2) has been disappointing =-=[2]-=-. As a consequence, solving MPCCs as NLPs has been commonly regarded as numerically unsafe. Recently, exciting new developments have demonstrated that the gloomy prognosis about the use of (1.2) may h... |

24 | QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints 13
- Jiang, Ralph
- 1999
(Show Context)
Citation Context ... lists the problem characteristics obtained with the scalar product formulation. The headings in Appendix B are explained in Table 2. The definition of the degree of degeneracy d1,d2,dm is taken from =-=[21]-=-. The corresponding columns refer to first-level degeneracy, d1, second-level degeneracy, d2, and mixed-degeneracy, dm. Table 2: Headings for tables in Appendix B. Heading Description name problem nam... |

21 | Benchmarking optimization software with COPS
- Dolan, Moré
- 2000
(Show Context)
Citation Context ...+ � 4σNR−1 σNR � (a − b) 2 + ab σNR Derivative values can be computed in a similarly stable fashion. 6.2 Performance Plots and Results . (6.1) Results are provided in two forms. The performance plots =-=[9]-=- in Figures 3 and 4 show the relative performance of each formulation in terms of iteration count and CPU time. These plots can be interpreted as follows. For every solver s and every problem p, the r... |

21 | Interior point algorithms, penalty methods and equilibrium problems
- Benson, Sen, et al.
- 2006
(Show Context)
Citation Context ...lty parameter can be found, and standard SQP methods converge. The convergence properties of interior point methods (IPMs) have also received renewed attention. Numerical experiments by Benson et al. =-=[BSSV03]-=- and by Raghunathan and Biegler [RB05] have shown that IPMs with minor modifications can be applied successfully to solve MPCCs. This practical success has encouraged theoretical studies of the conver... |

18 |
Global convergence of a trustregion SQP-filter algorithm for general nonlinear programming
- Toint, Wächter
(Show Context)
Citation Context ...SQP method promotes global convergence through the use of a filter. The filter accepts a trial point whenever the objective or the constraint violation is improved compared with all previous iterates =-=[14, 15, 18]-=-. 6.1 Preliminaries The solver includes an AMPL [19] interface that introduces slacks to formulate general complementarity constraints in the form (1.1) and handles the reformulation to the NLP (1.3) ... |

18 | The semismooth algorithm for large scale complementarity problems
- Munson, Facchinei, et al.
- 1999
(Show Context)
Citation Context ...pe of quadratic min-function (2.11) 2.0 While the number of parameters may appear unreasonably large, each formulation requires only three parameters to be set. The choice of λ = 0.7 also agrees with =-=[25]-=-, where λ = 0.8 is suggested. Note that since δ = 0.1, the Chen-Chen-Kanzow function is relaxed further. Care has to be taken when computing the smoothed natural residual function (2.9); it can be aff... |

18 |
Global convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints
- Anitescu
(Show Context)
Citation Context ...vated a closer investigation of the (local) convergence properties of SQP methods for MPCCs. Fletcher et al. [FLRS02] show that an SQP method converges locally to strongly stationary points. Anitescu =-=[Ani05]-=- establishes that an SQP method with elastic mode converges locally for MPCCs with (2). The key idea is to penalize X1x2 ≤ 0 and consider the resulting NLP, which satisfies MFCQ. Near a strongly stati... |

16 | Interior point methods for mathematical programs with complementarity constraints
- Raghunathan, Biegler
- 2005
(Show Context)
Citation Context ... SQP methods converge. The convergence properties of interior point methods (IPMs) have also received renewed attention. Numerical experiments by Benson et al. [BSSV03] and by Raghunathan and Biegler =-=[RB05]-=- have shown that IPMs with minor modifications can be applied successfully to solve MPCCs. This practical success has encouraged theoretical studies of the convergence properties of IPMs for MPCCs. Ra... |

15 | Generalized stationary points and an interior point method for mathematical programs with equilibrium constraints
- Liu, Sun
(Show Context)
Citation Context ...ical studies of the convergence properties of IPMs for MPCCs. Raghunathan and Biegler [27] relax x T 1 x2 ≤ 0 by a quantity proportional to the barrier parameter, which is driven to zero. Liu and Sun =-=[23]-=- propose a primal-dual IPM that also relaxes the complementarity constraint.sComplementarity Constraints as Nonlinear Equations 3 In this paper, we extend our results of [17] by considering NLP formul... |

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 ...mooth at the origin. We will show that this nonsmoothness does not affect the local convergence properties of the SQP method. The use of NCP functions for the solution of MPCCs has been considered in =-=[8, 10]-=-, where a sequence of smoothed NCP reformulation is solved. Our contribution is to show that this smoothing is not required. Thus we avoid the need to control the smoothing parameter that may be probl... |

13 |
A Newton–type method for positive semidefinite linear complementarity problems
- Fischer
- 1995
(Show Context)
Citation Context ... makes it consistent with strong stationarity, as will be shown later). It is not an NCP function, however, since φ BL(a,b) = 0 does not imply nonnegativity of a,b. 2. The Fischer-Burmeister function =-=[12]-=- is given by φ FB(a,b) = a + b − √ a 2 + b 2 . (2.5)s4 Sven Leyffer It is nondifferentiable at the origin, and its Hessian is unbounded at the origin. 3. The min-function [6] is the nonsmooth function... |

13 |
On the global convergence of a filter-SQP algorithm
- Toint
(Show Context)
Citation Context ... 1: Local SQP Algorithm for MPCCs In practice, we also include a globalization scheme to stabilize SQP. In our case, we use a filter [15] and a trust region to ensure convergence to stationary points =-=[18]-=-. The convergence theory of filter methods allows for three possible outcomes [18, Theorem 1]: (A) The algorithm terminates at a point that is locally infeasible. (B) The algorithm converges to a Kuhn... |

6 |
Numerical experience with solving MPECs as NLPs. Numerical Analysis Report NA/210
- Fletcher, Leyffer
- 2002
(Show Context)
Citation Context ...nosis about the use of (1.2) may have been premature. Standard sequential quadratic programming (SQP) solvers have been used to solve a large class of MPCCs, written as NLPs, reliably and efficiently =-=[16]-=-. This numerical success has motivated a closer investigation of the (local) convergence properties of SQP methods for MPCCs. In [17], it is shown that an SQP method converges locally to strongly stat... |

6 |
SNOPT: An SQP algorithm for large{ scale constrained optimization
- Gill, Murray, et al.
- 1997
(Show Context)
Citation Context ...and ν ∗ 2j > 0, ∀j ∈ D ∗ . [A5] The QP solver always chooses a linearly independent basis. We note that [A0] is readily implemented and that assumption [A5] holds for the QP solvers used within snopt =-=[20]-=- and filter [15]. The most restrictive assumptions are [A2] and [A3] because they exclude B-stationary points that are not strongly stationary. This fact is not surprising because it is well known tha... |

6 |
Barrier methods for mathematical programs with complementarity constraints (MPCCs
- Raghunathan, Biegler
- 2002
(Show Context)
Citation Context ...with minor modifications can be applied successfully to solve MPCCs. This practical success has encouraged theoretical studies of the convergence properties of IPMs for MPCCs. Raghunathan and Biegler =-=[27]-=- relax x T 1 x2 ≤ 0 by a quantity proportional to the barrier parameter, which is driven to zero. Liu and Sun [23] propose a primal-dual IPM that also relaxes the complementarity constraint.sComplemen... |

6 |
MPEC formulations and algorithms in process engineering
- Raghunathan, Biegler
- 2002
(Show Context)
Citation Context ...iently large penalty parameter can be found, and standard SQP methods converge. The convergence properties of interior point methods (IPMs) have also received renewed attention. Numerical experiments =-=[3, 28]-=- have shown that IPMs with minor modifications can be applied successfully to solve MPCCs. This practical success has encouraged theoretical studies of the convergence properties of IPMs for MPCCs. Ra... |

4 |
MacMPEC: AMPL collection of MPECs. Webpage
- Leyffer
- 2000
(Show Context)
Citation Context ...mplementarity constraints (MPCCs). Problems of this type arise in many engineering and economic applications; see the survey [11] and the monographs [24, 26]. The growing collections of test problems =-=[22, 7]-=- indicate that this an important area. MPCCs can be ∗ Preprint ANL/MCS-P1054-0603 1s2 Sven Leyffer expressed in general as minimize f(x) (1.1a) subject to cE(x) = 0 (1.1b) cI(x) ≥ 0 (1.1c) 0 ≤ x1 ⊥ x2... |

1 | MacMPEC: AMPL collection of MPECs. Webpage, [LPR96] [LS04] www.mcs.anl.gov/~leyffer/MacMPEC - Leyffer - 2000 |