## Interior methods for mathematical programs with complementarity constraints (2004)

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Venue: | SIAM J. Optim |

Citations: | 23 - 8 self |

### BibTeX

@ARTICLE{Leyffer04interiormethods,

author = {Sven Leyffer and Gabriel López-calva and Jorge Nocedal},

title = {Interior methods for mathematical programs with complementarity constraints},

journal = {SIAM J. Optim},

year = {2004},

volume = {17},

pages = {52--77}

}

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### Abstract

This paper studies theoretical and practical properties of interior-penalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then extended to an interior-relaxation approach. Superlinear convergence to strongly stationary points is also established. Two strategies for updating the penalty parameter are proposed, and their efficiency and robustness are studied on an extensive collection of test problems.

### Citations

374 | SNOPT: An SQP algorithm for large-scale constrained optimization
- Gill, Murray, et al.
- 2002
(Show Context)
Citation Context ...to the nonlinear programming formulation (1.3). Fletcher and Leyffer [11] tested loqo [25] and knitro [4] and observed that they were slower and less reliable than the SQP solvers filterSQP and snopt =-=[14]-=- (all codes as of 2002). This result contrasts starkly with the experience in nonlinear programming, where interior methods compete well with SQP methods. These studies have stimulated considerable in... |

264 | Benchmarking optimization software with performance profiles
- Dolan, Moré
(Show Context)
Citation Context ...ethods for solving MPCCs. In Figure 5 we report results for these four methods in terms of total number of iterations (indexed by j). The figures use the logarithmic performance profiles described in =-=[8]-=-. An important choice in the algorithms Classic and Dynamic is the initial value of π. In Figure 5(a) we show results for π 0 = 1, and in Figure 5(b) for π 0 = �∇f(x 0 )� (the latter rule is also used... |

179 | Nonlinear programming without a penalty function
- Fletcher, Leyer
(Show Context)
Citation Context ...no need to develop algorithms specifically for MPCCs. Numerical experiments by Fletcher and Leyffer [11] suggest that this goal is almost achieved by modern active-set SQP methods. In [11], filterSQP =-=[10]-=- was used to solve the problems in the MacMPEC collection [18], which contains over a hundred MPCCs, and fast convergence was almost always observed. The reason for this practical success is that, eve... |

169 | An interior point algorithm for nonconvex nonlinear programming
- Vanderbei, Shanno
- 1999
(Show Context)
Citation Context ...rate ways in which an SQP method may fail to converge. Interior methods are less successful when applied directly to the nonlinear programming formulation (1.3). Fletcher and Leyffer [11] tested loqo =-=[25]-=- and knitro [4] and observed that they were slower and less reliable than the SQP solvers filterSQP and snopt [14] (all codes as of 2002). This result contrasts starkly with the experience in nonlinea... |

113 |
Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity
- Scheel, Scholtes
(Show Context)
Citation Context ...i}, but we can write it as x T 1 x2 if we enforce the nonnegativity of x1, x2. This exact (2.1)s4 Sven Leyffer, Gabriel López-Calva, & Jorge Nocedal penalty reformulation of MPCCs has been studied in =-=[1, 2, 3, 17, 22, 23]-=-. Since problem (2.1) is smooth, we can safely apply standard nonlinear programming algorithms, such as interior methods, to solve it. The barrier problem associated to (2.1) is minimize f(x) + πx T 1... |

88 | Interior methods for nonlinear optimization
- Forsgren, Gill, et al.
- 2003
(Show Context)
Citation Context ...ed in stationarity for MPCCs is referred to [23]. 3.1 Global Convergence of the Interior-Penalty Algorithm Many algorithms have been proposed to solve the barrier problem in Step 2; see, for example, =-=[6, 13]-=- and the references therein. As is well known, these inner algorithms may fail to satisfy (2.7), and therefore Algorithm I can fail to complete Step 2. The analysis of the inner algorithm is beyond th... |

76 | An interior point algorithm for large scale nonlinear programming
- Byrd, Hribar, et al.
- 1999
(Show Context)
Citation Context ...ch an SQP method may fail to converge. Interior methods are less successful when applied directly to the nonlinear programming formulation (1.3). Fletcher and Leyffer [11] tested loqo [25] and knitro =-=[4]-=- and observed that they were slower and less reliable than the SQP solvers filterSQP and snopt [14] (all codes as of 2002). This result contrasts starkly with the experience in nonlinear programming, ... |

74 |
Convergence properties of a regularization scheme for mathematical programs with complementarity constraints
- SCHOLTES
(Show Context)
Citation Context ... in which (1.3e) is changed to x1ix2i ≤ θ, i = 1, ..., p, (1.4) and the relaxation parameter θ > 0 is driven to zero. This type of approach has been studied from a theoretical perspective by Scholtes =-=[24]-=- and Ralph and Wright [22]. Interior methods based on the relaxation (1.4) have been proposed by Liu and Sun [19] and Raghunathan and Biegler [21]. In both studies, the parameter θ is proportional to ... |

65 |
Trust-Region Methods
- Toint
- 2000
(Show Context)
Citation Context ...ed in stationarity for MPCCs is referred to [23]. 3.1 Global Convergence of the Interior-Penalty Algorithm Many algorithms have been proposed to solve the barrier problem in Step 2; see, for example, =-=[6, 13]-=- and the references therein. As is well known, these inner algorithms may fail to satisfy (2.7), and therefore Algorithm I can fail to complete Step 2. The analysis of the inner algorithm is beyond th... |

60 | Local convergence of SQP methods for mathematical programs with equilibrium constraints
- Fletcher, Leyffer, et al.
(Show Context)
Citation Context ... SQP solver is able to identify the right set of active constraints in the well-behaved program and converge to a solution. Failures, however, are still possible for the SQP approach. Fletcher et al. =-=[12]-=- give several examples that illustrate ways in which an SQP method may fail to converge. Interior methods are less successful when applied directly to the nonlinear programming formulation (1.3). Flet... |

34 | On solving mathematical programs with complementarity constraints as nonlinear programs
- Anitescu
- 2001
(Show Context)
Citation Context ...ation of a single penalty function. The appropriate value of π is, however, unknown in advance and must be estimated during the course of the minimization. This approach was first studied by Anitescu =-=[1]-=- in the context of active-set SQP methods, although it had been used before to solve engineering problems (see, e.g., [9]). It has been adopted as a heuristic to solve MPCCs with interior methods in l... |

31 | An interior algorithm for nonlinear optimization that combines line search and trust region steps
- Waltz, Morales, et al.
(Show Context)
Citation Context ...hm Dynamic. x (j−m+1) � 2 , (5.1) We implemented these two algorithms as an extension of our matlab solver ipm-d. This solver is based on the interior algorithm for nonlinear programming described in =-=[26]-=-, with one change: ipm-d handles negative curvature by adding a multiple of the identity to the Hessian of the Lagrangian, as in [25], instead of switching to conjugate-gradient iterations. We chose t... |

30 | Convergence of a penalty method for mathematical programming with complementarity constraints
- Hu, Ralph
- 2000
(Show Context)
Citation Context ...olve MPCCs with interior methods in loqo by Benson et al. [3], who present very good numerical results on the MacMPEC set. A more general class of exact penalty functions was analyzed by Hu and Ralph =-=[17]-=-, who derive global convergence results for a sequence of penalty problems that are solved exactly. Anitescu [2] derives similar global results in the context of inexact subproblem solves. In this pap... |

29 |
Solving mathematical program with complementarity constraints as nonlinear programs
- Fletcher, Leyffer
- 2004
(Show Context)
Citation Context ...f MPCCs. If this level of robustness could be attained (and this is a laudable goal) there might be no need to develop algorithms specifically for MPCCs. Numerical experiments by Fletcher and Leyffer =-=[11]-=- suggest that this goal is almost achieved by modern active-set SQP methods. In [11], filterSQP [10] was used to solve the problems in the MacMPEC collection [18], which contains over a hundred MPCCs,... |

26 | On the local behavior of an interior point method for nonlinear programming
- Byrd, Liu, et al.
- 1997
(Show Context)
Citation Context ..., in addition, z is close to z ∗ , then L1µ ≤ �z ∗ − z ∗ (µ)� ≤ U1µ, (4.12) L2�Fµ(z; π)� ≤ �z − z ∗ (µ)� ≤ U2�Fµ(z; π)�. (4.13) (Condition (4.12) is Corollary 3.14 in [13], and (4.13) is Lemma 2.4 in =-=[5]-=-.) Here and in the rest of the proof Li and Ui denote positive constants; recall that � · � denotes the infinity norm (without loss of generality). By standard Newton analysis (see, e.g., Theorem 2.3 ... |

26 | Some properties of regularization and penalization schemes for MPECs
- Ralph, Wright
- 2004
(Show Context)
Citation Context ...d to x1ix2i ≤ θ, i = 1, ..., p, (1.4) and the relaxation parameter θ > 0 is driven to zero. This type of approach has been studied from a theoretical perspective by Scholtes [24] and Ralph and Wright =-=[22]-=-. Interior methods based on the relaxation (1.4) have been proposed by Liu and Sun [19] and Raghunathan and Biegler [21]. In both studies, the parameter θ is proportional to the barrier parameter µ an... |

21 | Interior point algorithms, penalty methods and equilibrium problems
- Benson, Sen, et al.
- 2006
(Show Context)
Citation Context ...tive-set SQP methods, although it had been used before to solve engineering problems (see, e.g., [9]). It has been adopted as a heuristic to solve MPCCs with interior methods in loqo by Benson et al. =-=[3]-=-, who present very good numerical results on the MacMPEC set. A more general class of exact penalty functions was analyzed by Hu and Ralph [17], who derive global convergence results for a sequence of... |

18 |
Global convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints
- Anitescu
(Show Context)
Citation Context ...MPEC set. A more general class of exact penalty functions was analyzed by Hu and Ralph [17], who derive global convergence results for a sequence of penalty problems that are solved exactly. Anitescu =-=[2]-=- derives similar global results in the context of inexact subproblem solves. In this paper, we focus on the penalization approach, because we view it as a general tool for handling degenerate nonlinea... |

16 | Interior point methods for mathematical programs with complementarity constraints
- Raghunathan, Biegler
- 2005
(Show Context)
Citation Context ...en studied from a theoretical perspective by Scholtes [24] and Ralph and Wright [22]. Interior methods based on the relaxation (1.4) have been proposed by Liu and Sun [19] and Raghunathan and Biegler =-=[21]-=-. In both studies, the parameter θ is proportional to the barrier parameter µ and is updated only at the end of each barrier problem. Raghunathan and Biegler focus on local analysis and report very go... |

15 | Generalized stationary points and an interior point method for mathematical programs with equilibrium constraints
- Liu, Sun
(Show Context)
Citation Context ...ero. This type of approach has been studied from a theoretical perspective by Scholtes [24] and Ralph and Wright [22]. Interior methods based on the relaxation (1.4) have been proposed by Liu and Sun =-=[19]-=- and Raghunathan and Biegler [21]. In both studies, the parameter θ is proportional to the barrier parameter µ and is updated only at the end of each barrier problem. Raghunathan and Biegler focus on ... |

13 |
Interior-Point ℓ2 penalty methods for nonlinear programming with strong global convergence properties, CORC
- Chen, Goldfarb
- 2005
(Show Context)
Citation Context ...ese point to the need for more general regularization schemes for interior methods that can cope with both MPCCs and with other forms of degeneracy. This topic is the subject of current investigation =-=[15, 16]-=-. 6 Conclusions Interior methods can be an efficient and robust tool for solving MPCCs, when appropriately combined with a regularization scheme. In this article, we have studied an interior-penalty a... |

9 |
An interior-point l1-penalty method for nonlinear optimization
- Toint
- 2003
(Show Context)
Citation Context ...ese point to the need for more general regularization schemes for interior methods that can cope with both MPCCs and with other forms of degeneracy. This topic is the subject of current investigation =-=[15, 16]-=-. 6 Conclusions Interior methods can be an efficient and robust tool for solving MPCCs, when appropriately combined with a regularization scheme. In this article, we have studied an interior-penalty a... |

7 | A two-sided relaxation scheme for mathematical programs with equilibrium constraints
- DeMiguel, Friedlander, et al.
(Show Context)
Citation Context ...ts. Numerical difficulties may arise when the relaxation parameter gets small, since the interior of the regularized problem shrinks toward the empty set.sInterior Methods for MPCCs 3 DeMiguel et al. =-=[7]-=- address this problem by proposing a different relaxation scheme where, in addition to (1.4), the nonnegativity bounds on the variables are relaxed to x1i ≥ −δ, x2i ≥ −δ. (1.5) Under fairly general as... |

7 | An interior-point method for MPECS based on strictly feasible relaxations
- Miguel, Friedlander, et al.
- 2005
(Show Context)
Citation Context ...ts. Numerical difficulties may arise when the relaxation parameter gets small, since the interior of the regularized problem shrinks toward the empty set.sInterior Methods for MPCCs 3 DeMiguel et al. =-=[8]-=- address this problem by proposing a different relaxation scheme where, in addition to (1.4), the nonnegativity bounds on the variables are relaxed to x1i ≥ −δ, x2i ≥ −δ. (1.5) Under fairly general as... |

6 |
Barrier methods for mathematical programs with complementarity constraints (MPCCs
- Raghunathan, Biegler
- 2002
(Show Context)
Citation Context ...een studied from a theoretical perspective by Scholtes [23] and Ralph and Wright [7]. Interior methods based on the relaxation (1.4) have been proposed by Liu and Sun [18] and Raghunathan and Biegler =-=[21]-=-. In both studies, the parameter θ is proportional to the barrier parameter µ and is updated only at the end of each barrier problem. Raghunathan and Biegler focus on local analysis and report very go... |

4 |
An interior point ℓ1 penalty method for nonlinear optimization
- Toint
- 2003
(Show Context)
Citation Context ...ese point to the need for more general regularization schemes for interior methods that can cope with both MPCCs and with other forms of degeneracy. This topic is the subject of current investigation =-=[14, 15]-=-.s26 Sven Leyffer, Gabriel López-Calva, & Jorge Nocedal 6 Conclusions Interior methods can be an efficient and robust tool for solving MPCCs, when appropriately combined with a regularization scheme. ... |

3 |
On the solution of a minimum weight elastoplastic problem involving displacement and complementarity constraints, Computer Methods in Applied Mechanics and Engineering (in print
- Ferris, Tin-Loi
- 1999
(Show Context)
Citation Context ...ng the course of the minimization. This approach was first studied by Anitescu [1] in the context of active-set SQP methods, although it had been used before to solve engineering problems (see, e.g., =-=[9]-=-). It has been adopted as a heuristic to solve MPCCs with interior methods in loqo by Benson et al. [3], who present very good numerical results on the MacMPEC set. A more general class of exact penal... |

1 | MacMPEC: AMPL collection of MPECs. Web - Leyffer - 2000 |

1 | Regularization for nonlinear programming via exact-penalty methods - López-Calva - 2005 |

1 | Algorithms for Nonlinear Optimization: Problems with Complementarity Constraints and Derivative-Free Methods - López-Calva - 2004 |