## Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties (2005)

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Venue: | Math. Program |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Anitescu05elastic-modealgorithms,

author = {Mihai Anitescu and Paul Tseng and Stephen J. Wright and Stationarity Properties},

title = {Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties},

journal = {Math. Program},

year = {2005},

volume = {110},

pages = {337--371}

}

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### Abstract

Abstract. The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first- or second-order point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, M-stationarity, or C-stationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic-mode formulation is solved inexactly. Key words. Nonlinear programming, equilibrium constraints, complementarity constraints, elastic-mode formulation, strong stationarity, C-stationarity, Mstationarity. AMS subject classifications 49M30, 49M37, 65K05, 90C30, 90C33 1.

### Citations

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(Show Context)
Citation Context ...way to assert that MPEC-SLICQ holds “almost always” or “generically.” That is, can we define a function space containing the problem data, and a topology in the space (typically, the Whitney topology =-=[9]-=-), and then argue that the property in question holds for an open and dense subset in the data space? In this vein, Jongen, Jonker, and Twilt [11,12] have shown that LICQ holds “almost always” for non... |

161 | Nonlinear programming without a penalty function
- Fletcher, Leyffer
- 2002
(Show Context)
Citation Context ...solved by methods that enforce exact complementarity between the constraints G T x ≥ 0 and H T x ≥ 0 and their respective multipliers at each iteration (including active-set methods such as filterSQP =-=[4]-=-), the generated point x k will become exactly feasible with respect to the complementarity constraints G T x ⊥ H T x, once ck exceeds a certain threshold. Theorem 5. Let {ck}, {ɛk}, and {δk} be nonne... |

107 |
Nonsmooth Approach to Optimization Problems with Equilibrium Constraints
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- 1998
(Show Context)
Citation Context ...n obstacle. The membrane is attached to a support whose shape is parametrized by the variables in the optimization problem. Descriptions of all our problems can be found in Outrata, Kocvara, and Zowe =-=[19]-=-. We use the AMPL formulations from the MacMPEC Library of Leyffer [14]; see also Fletcher and Leyffer [5]. – Incidence set identification [19, Section 9.4]. This MPEC seeks the shape of the support s... |

99 |
Mathematical Programs with Complementarity Constraints: Stationarity
- Scheel, Scholtes
- 2000
(Show Context)
Citation Context ...ons, certain necessary optimality conditions for the MPEC (1). Different types of necessary optimality conditions have been developed, the strongest and most desirable of which is strong stationarity =-=[23]-=-; see Definition 1 below. Under MPEC-LICQ (see Definition 2), strong stationarity is equivalent to the notion of B-stationarity [6]. Two weaker conditions, M-stationarity and C-stationarity [18,23], w... |

60 |
Convergence properties of a regularization scheme for mathematical programs with complementarity constraints
- Scholtes
- 2001
(Show Context)
Citation Context ...y is equivalent to the notion of B-stationarity [6]. Two weaker conditions, M-stationarity and C-stationarity [18,23], will also be of interest (see Definition 3). A regularization method of Scholtes =-=[24]-=- achieves M-stationarity under MPECLICQ and achieves strong stationarity under an additional upper-level strict complementarity (ULSC) condition. A relaxation method of Lin and Fukushima [15] and a pe... |

43 |
Generalized equations and their solutions, part ii: Applications to nonlinear programming
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- 1982
(Show Context)
Citation Context ...e of second-order points of Reg(tk) may approach an M-stationary point of (1) while Algorithm ElasticExact terminates finitely at a strongly stationary point of (1). It is constructed from Robinson’s =-=[22]-=- example of a strict local solution that is not isolated. Example 2. Consider the function F : IR → IR defined as follows: F (y) def = �y 0 t 6 sin(1/t) dt. It is easy to verify that F is three times ... |

27 |
Convergence of a smoothing continuation method for mathematical programs with complementarity constraints
- Fukushima, Pang
- 1999
(Show Context)
Citation Context ...d, the strongest and most desirable of which is strong stationarity [23]; see Definition 1 below. Under MPEC-LICQ (see Definition 2), strong stationarity is equivalent to the notion of B-stationarity =-=[6]-=-. Two weaker conditions, M-stationarity and C-stationarity [18,23], will also be of interest (see Definition 3). A regularization method of Scholtes [24] achieves M-stationarity under MPECLICQ and ach... |

25 |
Numerical Computing with
- Overton
- 2001
(Show Context)
Citation Context ... from the fact that λk i ≥ 0 and gi(xk ) + ζk ≥ −ɛk for all i (due to the fourth row of (10)). Since i /∈ Ig ⇒ gi(xk ) + ζk ≥ gi(xk ) ≥ ρ > 0, it follows that ρ � λ k � i ≤ ɛk + ɛk λ k i , for all k. =-=(20)-=- Then When � i∈Ig λk i i/∈Ig p� λ k i ∇gi(x k ) = � i=1 i∈IH i∈Ig ≥ 1, we have immediately from (20) that i∈Ig λ k i � � i/∈Ig λki � i∈Ig λki ∇gi(x k ) + i∈Ig ≤ 2ɛk . (21) ρ � j /∈Ig λkj ∇gj(xk ) � j∈... |

23 |
Convergence of a penalty method for mathematical programs with complementarity constraints
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- 2004
(Show Context)
Citation Context ...r MPECLICQ and achieves strong stationarity under an additional upper-level strict complementarity (ULSC) condition. A relaxation method of Lin and Fukushima [15] and a penalty method of Hu and Ralph =-=[10]-=-, penalizing the complementarity constraint, have similar global convergence properties. A smoothing method of Fukushima and Pang [6] achieves strong stationarity under MPEC-LICQ and an additional asy... |

22 |
2001. How stringent is the linear independence assumption for mathematical programs with stationarity constraints
- S, Stöhr
(Show Context)
Citation Context ...uestion holds for an open and dense subset in the data space? In this vein, Jongen, Jonker, and Twilt [11,12] have shown that LICQ holds “almost always” for nonlinear programs, and Scholtes and Stöhr =-=[25]-=- have shown that MPEC-LICQ holds “almost always” for MPEC. The corresponding conjecture for MPEC-SLICQ, for the same function space and topology as the one used in [25] would be as follows: The set of... |

21 | 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 ...tion of a sequence of points satisfying exactly certain second-order necessary optimality conditions. Only in the case of linear constraints has a practical method been developed (Fukushima and Tseng =-=[7]-=-). We are led to ask: Can global convergence (to C- or M- or strongly stationarity points) be achieved under weaker assumptions or for more practical methods? In this paper, we study this question for... |

21 |
Optimality conditions for a class of mathematical programs with equilibrium constraints
- Outrata
- 1999
(Show Context)
Citation Context ...arity [23]; see Definition 1 below. Under MPEC-LICQ (see Definition 2), strong stationarity is equivalent to the notion of B-stationarity [6]. Two weaker conditions, M-stationarity and C-stationarity =-=[18,23]-=-, will also be of interest (see Definition 3). A regularization method of Scholtes [24] achieves M-stationarity under MPECLICQ and achieves strong stationarity under an additional upper-level strict c... |

21 | Some properties of regularization and penalization schemes for MPECs
- Wright
(Show Context)
Citation Context ...components. A similar formulation was studied by Anitescu [1,2], while a variant with ζ fixed at zero was investigated (2)sGlobal Convergence of Elastic-Mode Algorithms for MPEC 3 by Ralph and Wright =-=[21]-=-. The penalty method in [10] is based on this variant. Our analysis may also be extended to this variant, as well as to a mixed variant whereby ζ is fixed at zero for a subset of the constraints (see ... |

14 | Generalized stationary points and an interior point method for mathematical programs with equilibrium constraints
- Liu, Sun
- 2002
(Show Context)
Citation Context ...at � i∈IH\IG ⎧ ⎨ Hi + ⎩ HT i xk GT i xk Gi, for i ∈ IH\IG , Hi, for i ∈ IG ∩ IH, � k (νi − ckG T i x k )Hi − ck(H T i x k � � )Gi = i∈IH\IG Since {H T i xk /G T i xk } → 0 for i ∈ IH\IG, we have from =-=(16)-=- that { ˜ H k i } → Hi, for i ∈ IH. (16) (ν k i − ckG T i x k ) ˜ H k i + O(ɛk). (17) A similar definition of ˜ G k i for i ∈ IG yields for the second-to-last summation in (14) that � i∈IG\IH � k (τi ... |

13 |
On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints. Preprint ANL/MCS-P864-1200.To appear in
- Anitescu
(Show Context)
Citation Context ... x) T (H T x) subject to g(x) ≥ −ζep, ζeq ≥ h(x) ≥ −ζeq, 0 ≤ ζ ≤ ¯ ζ, G T x ≥ 0, H T x ≥ 0, where el is the vector (1, 1, . . . , 1) T with l components. A similar formulation was studied by Anitescu =-=[1,2]-=-, while a variant with ζ fixed at zero was investigated (2)sGlobal Convergence of Elastic-Mode Algorithms for MPEC 3 by Ralph and Wright [21]. The penalty method in [10] is based on this variant. Our ... |

12 |
Critical sets in parametric optimization
- Jongen, Jonker, et al.
- 1986
(Show Context)
Citation Context ...topology in the space (typically, the Whitney topology [9]), and then argue that the property in question holds for an open and dense subset in the data space? In this vein, Jongen, Jonker, and Twilt =-=[11,12]-=- have shown that LICQ holds “almost always” for nonlinear programs, and Scholtes and Stöhr [25] have shown that MPEC-LICQ holds “almost always” for MPEC. The corresponding conjecture for MPEC-SLICQ, f... |

12 | Piecewise sequential quadratic programming for mathematical programs with nonlinear complementarity constraints
- Luo, Pang, et al.
- 1998
(Show Context)
Citation Context ... of (1) if the following set of vectors is linearly independent: K def = {∇gi(x ∗ )}i∈Ig ∪ {∇hi(x ∗ )}i=1,2,...,q ∪ {Gi}i∈IG ∪ {Hi}i∈IH . (7) The following result, dating back to Luo, Pang, and Ralph =-=[17]-=- but stated here in the form of Scheel and Scholtes [23, Theorem 2], shows that, under MPEC-LICQ, strong stationarity is a set of (first-order) necessary optimality conditions for the MPEC. Theorem 1.... |

10 |
New relaxation method for mathematical programs with complementarity constraints
- Lin, Fukushima
(Show Context)
Citation Context ...Scholtes [24] achieves M-stationarity under MPECLICQ and achieves strong stationarity under an additional upper-level strict complementarity (ULSC) condition. A relaxation method of Lin and Fukushima =-=[15]-=- and a penalty method of Hu and Ralph [10], penalizing the complementarity constraint, have similar global convergence properties. A smoothing method of Fukushima and Pang [6] achieves strong stationa... |

9 |
MacMPEC AMPL collection of mathematical programs with equilibrium constraints
- Leyffer
(Show Context)
Citation Context ...trized by the variables in the optimization problem. Descriptions of all our problems can be found in Outrata, Kocvara, and Zowe [19]. We use the AMPL formulations from the MacMPEC Library of Leyffer =-=[14]-=-; see also Fletcher and Leyffer [5]. – Incidence set identification [19, Section 9.4]. This MPEC seeks the shape of the support so that the contact region is as close as possible to a prescribed shape... |

6 | An interior-point method for MPECs based on strictly feasible relaxations - DeMiguel, Friedlander, et al. - 2004 |

6 |
Optimization problems with equilibrium constraints and their numerical solution
- Kočvara, Outrata
(Show Context)
Citation Context ...k , ν k , π −k , π +k ) be multipliers associated with (x k , ζk) (from (10)). From the final row of (10), we have that, for all k, ν k i (H T i x k ) ≤ (ν k ) T (H T x k ) ≤ ɛk, i = 1, 2, . . . , m, =-=(13)-=- so for i /∈ IH, since HT i xk is bounded away from zero, we have that νk i = O(ɛk). By similar reasoning, we have that τ k i = O(ɛk) for i /∈ IG. Using these two facts, we can write the first row of ... |

2 |
experience with solving MPECs as NLPs
- Numerical
- 2002
(Show Context)
Citation Context ...ization problem. Descriptions of all our problems can be found in Outrata, Kocvara, and Zowe [19]. We use the AMPL formulations from the MacMPEC Library of Leyffer [14]; see also Fletcher and Leyffer =-=[5]-=-. – Incidence set identification [19, Section 9.4]. This MPEC seeks the shape of the support so that the contact region is as close as possible to a prescribed shape. The objective function is a measu... |

1 |
convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints
- Global
(Show Context)
Citation Context ... x) T (H T x) subject to g(x) ≥ −ζep, ζeq ≥ h(x) ≥ −ζeq, 0 ≤ ζ ≤ ¯ ζ, G T x ≥ 0, H T x ≥ 0, where el is the vector (1, 1, . . . , 1) T with l components. A similar formulation was studied by Anitescu =-=[1,2]-=-, while a variant with ζ fixed at zero was investigated (2)sGlobal Convergence of Elastic-Mode Algorithms for MPEC 3 by Ralph and Wright [21]. The penalty method in [10] is based on this variant. Our ... |

1 |
Hooking your solver to AMPL, technical report
- Gay
- 1993
(Show Context)
Citation Context ...terSQP the primal variables (x k , ζk), the Lagrange multipliers (λ k , µ −k , µ +k , τ k , ν k ), the constraint gradients, and the Hessian of the Lagrangian for the last subproblem PF(ck). (See Gay =-=[8]-=- for details on access to AMPL structures by functions defined in Matlab.) We then compute the active sets at x k , using a tolerance of δ = 10 −6 , as follows: Ig IG IH def = {i ∈ {1, 2, . . . , p} |... |

1 |
families of optimization problems: Equality constraints
- One-parameter
- 1986
(Show Context)
Citation Context ...topology in the space (typically, the Whitney topology [9]), and then argue that the property in question holds for an open and dense subset in the data space? In this vein, Jongen, Jonker, and Twilt =-=[11,12]-=- have shown that LICQ holds “almost always” for nonlinear programs, and Scholtes and Stöhr [25] have shown that MPEC-LICQ holds “almost always” for MPEC. The corresponding conjecture for MPEC-SLICQ, f... |