## Some properties of regularization and penalization schemes for MPECs (2004)

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Venue: | Optimization Methods and Software |

Citations: | 28 - 2 self |

### BibTeX

@ARTICLE{Ralph04someproperties,

author = {Daniel Ralph and Stephen and J. Wright},

title = {Some properties of regularization and penalization schemes for MPECs},

journal = {Optimization Methods and Software},

year = {2004},

volume = {19},

pages = {527--556}

}

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### Abstract

Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. Existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.

### Citations

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Perturbation Analysis of Optimization Problems
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Citation Context ...s formulation, but with the O(t 1/2 ) estimate replaced by O(t 1/4 ). (The proof technique is quite different; unlike the proofs in Section 3.1, it does not rely on the results of Bonnans and Shapiro =-=[3]-=-.) Finally, in Section 5, we discuss a nonlinear programming reformulation based on the exact ℓ1 penalty function. For a given nonnegative parameter ρ, this reformulation is as follows: (1.5) PF(ρ) : ... |

203 |
Mathematical Programs with Equilibrium Constraints
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Citation Context ...Gi(x∗)T s = 0 for all i with τ∗i 6= 0} ∩ {s |∇Hi(x∗)T s = 0 for all i with ν∗i 6= 0}. For the MPECs, the critical directions may be different for every “branch” of the feasible set containing x∗; see =-=[20]-=- for various “piecewise” optimality conditions using this motivation. For any partition I ∪ J of J0G ∩ J0H , let T ∗(I, J) def= T̄ ∩{s |∇Gi(x∗)T s = 0 for all i ∈ I}∩{s |∇Hi(x∗)T s = 0 for all i ∈ J}.... |

121 | 2000), ‘Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity - Scheel, Scholtes |

96 | Interior-point methods for nonconvex nonlinear programming: Filter-methods and merit functions
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Citation Context ...t al. [19] and Fletcher et al. [7]. Interior-point methods have been proposed by de Miguel, Friedlander, Nogales and Scholtes [5] and Raghunathan and Biegler [23], while Benson, Shanno, and Vanderbei =-=[2]-=- have performed a computational study involving the LOQO interior-point code and the MacMPEC test set (Leyffer [17]). An anonymous referee has alerted us to a forthcoming paper by Izmailov [14]. We do... |

75 |
Convergence properties of regularization schemes for mathematical programs with complementarity constraints
- Scholtes
- 2001
(Show Context)
Citation Context ...ns (for example, replacing G(x) ⊥ H(x) by G(x) T H(x) = 0) do not satisfy constraint qualifications [4, 29] at any feasible point. In this paper, we study a regularization scheme analyzed by Scholtes =-=[28]-=- in which (1.1) is approximated by the following nonlinear program, which is parametrized by the nonnegative scalar t: Reg(t) : minx f(x) subject to (1.2) g(x) ≥ 0, h(x) = 0, G(x) ≥ 0, H(x) ≥ 0, Gi(x)... |

65 | A smoothing method for mathematical programs with equilibrium constraints
- Facchinei, Jiang, et al.
- 1999
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Citation Context ...tion schemes, there are several other nonlinear programming approached to (1.1) with similar motivations, starting with Fukushima and Pang’s analysis [8] of the smoothing scheme of Facchinei 2set al. =-=[6]-=-, and including the penalty approaches analyzed by Hu and Ralph [12] and Huang, Yang, and Zhu [13]. Lin and Fukushima [18] have studied the issue of identifying active constraints in smoothing, regula... |

61 | Local convergence of SQP methods for mathematical programs with equilibrium constraints
- Fletcher, Leyffer, et al.
- 2006
(Show Context)
Citation Context ...Tseng and Fukushima [9], who use an ɛ-active set method; and Zhang and Liu [30], who use an extreme-ray descent method. SQP-based methods for MPECs can be found in Liu et al. [19] and Fletcher et al. =-=[7]-=-. Interior-point methods have been proposed by de Miguel, Friedlander, Nogales and Scholtes [5] and Raghunathan and Biegler [23], while Benson, Shanno, and Vanderbei [2] have performed a computational... |

47 |
Generalized equations and their solutions. Part II: Applications to nonlinear programming
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- 1982
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Citation Context ...r.) Note that if we assume RNLP-SOSC rather than the less stringent MPEC-SOSC, the quadratic increase result becomes a trivial consequence of standard nonlinear programming theory; see again Robinson =-=[26]-=-. 3. Properties of Solutions of Reg(t). In this section, we investigate the minimizers of Reg(t) for small values of t. Our starting point is a result of Scholtes [28, Theorem 4.1], which we state in ... |

43 |
The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions
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Citation Context ...etical interest, which arises from the fact that their most natural nonlinear programming formulations (for example, replacing G(x) ⊥ H(x) by G(x) T H(x) = 0) do not satisfy constraint qualifications =-=[4, 29]-=- at any feasible point. In this paper, we study a regularization scheme analyzed by Scholtes [28] in which (1.1) is approximated by the following nonlinear program, which is parametrized by the nonneg... |

36 |
A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming
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- 1977
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Citation Context ... T d > 0, i ∈ IG ∩ IH; and ∇Gi(x ∗ ), i ∈ IG \ IH, ∇Hi(x ∗ ), i ∈ IH \ IG, ∇hi(x ∗ ), i = 1, 2, . . . , q are all linearly independent. 5s(It is easy to show, by using an argument like that of Gauvin =-=[10]-=- for nonlinear programming, that MPEC-MFCQ holds if and only if the set of multipliers (λ∗ , µ ∗ , τ ∗ , ν∗ ) satisfying (2.5) is bounded.) We now define three varieties of strict complementarity at a... |

35 |
Convergence of a smoothing continuation method for mathematical programs with complementarity constraints
- Fukushima, Pang
- 1998
(Show Context)
Citation Context ...per focuses on certain regularization and penalization schemes, there are several other nonlinear programming approached to (1.1) with similar motivations, starting with Fukushima and Pang’s analysis =-=[8]-=- of the smoothing scheme of Facchinei 2set al. [6], and including the penalty approaches analyzed by Hu and Ralph [12] and Huang, Yang, and Zhu [13]. Lin and Fukushima [18] have studied the issue of i... |

32 | Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems - Robinson - 1976 |

31 | Convergence of a Penalty Method of Mathematical Programming with Complementarity Constraints
- Hu, Ralph
- 2004
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Citation Context ...hed to (1.1) with similar motivations, starting with Fukushima and Pang’s analysis [8] of the smoothing scheme of Facchinei 2set al. [6], and including the penalty approaches analyzed by Hu and Ralph =-=[12]-=- and Huang, Yang, and Zhu [13]. Lin and Fukushima [18] have studied the issue of identifying active constraints in smoothing, regularization, and penalty methods. More recently, Anitescu [1] has studi... |

24 | QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constaints
- Jiang, Ralph
- 1999
(Show Context)
Citation Context ...ch recognize the disjunctive nature of MPEC constraints are well studied. We mention the globally convergent methods for MPECs with linear constraint functions proposed or analyzed by Jiang and Ralph =-=[15]-=- (see [20, Chapter 6] and [16] for local convergence analysis); Tseng and Fukushima [9], who use an ɛ-active set method; and Zhang and Liu [30], who use an extreme-ray descent method. SQP-based method... |

24 | Exact penalization and necessary optimality conditions for generalized bilevel programming problems
- Ye, Zhu, et al.
- 1997
(Show Context)
Citation Context ...oretical interest, which arises from the fact that their most natural nonlinear programming formulations (for example, replacing G(x) ⊥ H(x) by G(x)TH(x) = 0) do not satisfy constraint qualifications =-=[4, 29]-=- at any feasible point. In this paper, we study a regularization scheme analyzed by Scholtes [28] in which (1.1) is approximated by the following nonlinear program, which is parameterized by the nonne... |

23 | An implementable active-set algorithm for computing a b-stationary point of a mathematical program with linear complementarity constraints
- Fukushima, Tseng
- 2002
(Show Context)
Citation Context ... globally convergent methods for MPECs with linear constraint functions proposed or analyzed by Jiang and Ralph [15] (see [20, Chapter 6] and [16] for local convergence analysis); Tseng and Fukushima =-=[9]-=-, who use an ɛ-active set method; and Zhang and Liu [30], who use an extreme-ray descent method. SQP-based methods for MPECs can be found in Liu et al. [19] and Fletcher et al. [7]. Interior-point met... |

20 |
First- and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems
- Maurer, Zowe
- 1979
(Show Context)
Citation Context ... (2.13) � ≤ r0. Proof. This result follows from Scheel and Scholtes [27, Theorem 7(2)] and basic theory concerning quadratic growth for standard nonlinear programming; see for example Maurer and Zowe =-=[22]-=- and Robinson [26, Theorem 2.2]. We can still prove quadratic increase if we drop the strong stationarity assumption, and assume instead B-stationarity of x ∗ along with an SOSC for all nonlinear prog... |

19 | Interior point methods for Mathematical Programs with Complementarity Constraints (MPCCs
- Raghunathan, Biegler
- 2005
(Show Context)
Citation Context ...ed methods for MPECs can be found in Liu et al. [19] and Fletcher et al. [7]. Interior-point methods have been proposed by de Miguel, Friedlander, Nogales and Scholtes [5] and Raghunathan and Biegler =-=[23]-=-, while Benson, Shanno, and Vanderbei [2] have performed a computational study involving the LOQO interior-point code and the MacMPEC test set (Leyffer [17]). An anonymous referee has alerted us to a ... |

17 | Piecewise sequential quadratic programming for mathematical programs with complementarity constraints
- Luo, Pang, et al.
- 1998
(Show Context)
Citation Context ...s, the linear independence constraint qualification (LICQ) is satisfied for the RNLP (2.4). We have the following result concerning first-order necessary conditions dating back to Luo, Pang and Ralph =-=[21]-=- but stated in the form of Scheel and Scholtes [27, Theorem 2]. Theorem 2.4. Suppose that x ∗ is a local minimizer of (1.1). Then if the MPECLICQ condition holds at x ∗ , then x ∗ is strongly stationa... |

16 |
On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints
- Anitescu
- 2000
(Show Context)
Citation Context ...nd Ralph [12] and Huang, Yang, and Zhu [13]. Lin and Fukushima [18] have studied the issue of identifying active constraints in smoothing, regularization, and penalty methods. More recently, Anitescu =-=[1]-=- has studied the “elastic mode” for nonlinear programming, in conjunction with a sequential quadratic programming (SQP) algorithm, and focuses particularly on MPECs. Anitescu’s formulation is similar ... |

15 |
Exact penalization and necessary optimality conditions for generalized bilevel programming problems
- Ye, Zhu, et al.
- 1997
(Show Context)
Citation Context ...etical interest, which arises from the fact that their most natural nonlinear programming formulations (for example, replacing G(x) ⊥ H(x) by G(x) T H(x) = 0) do not satisfy constraint qualifications =-=[4, 29]-=- at any feasible point. In this paper, we study a regularization scheme analyzed by Scholtes [28] in which (1.1) is approximated by the following nonlinear program, which is parametrized by the nonneg... |

9 |
A sequential smooth penalization approach to mathematical programs with complementarity constraints
- Huang, Yang, et al.
- 2001
(Show Context)
Citation Context ...vations, starting with Fukushima and Pang’s analysis [8] of the smoothing scheme of Facchinei 2set al. [6], and including the penalty approaches analyzed by Hu and Ralph [12] and Huang, Yang, and Zhu =-=[13]-=-. Lin and Fukushima [18] have studied the issue of identifying active constraints in smoothing, regularization, and penalty methods. More recently, Anitescu [1] has studied the “elastic mode” for nonl... |

9 |
MacMPEC AMPL collection of mathematical programs with equilibrium constraints
- Leyffer
- 2000
(Show Context)
Citation Context ...nd Scholtes [5] and Raghunathan and Biegler [23], while Benson, Shanno, and Vanderbei [2] have performed a computational study involving the LOQO interior-point code and the MacMPEC test set (Leyffer =-=[17]-=-). An anonymous referee has alerted us to a forthcoming paper by Izmailov [14]. We do not have access to an English translation of this paper, but believe that it includes analysis similar to some of ... |

6 |
Extension of quasi-Newton methods to mathematical programs with complementarity constraints
- Jiang, Ralph
(Show Context)
Citation Context ...ature of MPEC constraints are well studied. We mention the globally convergent methods for MPECs with linear constraint functions proposed or analyzed by Jiang and Ralph [15] (see [20, Chapter 6] and =-=[16]-=- for local convergence analysis); Tseng and Fukushima [9], who use an ɛ-active set method; and Zhang and Liu [30], who use an extreme-ray descent method. SQP-based methods for MPECs can be found in Li... |

6 | A robust SQP methods for mathematical programs with complementarity constraints
- Liu, Perakis, et al.
- 2003
(Show Context)
Citation Context ...l convergence analysis); Tseng and Fukushima [9], who use an ɛ-active set method; and Zhang and Liu [30], who use an extreme-ray descent method. SQP-based methods for MPECs can be found in Liu et al. =-=[19]-=- and Fletcher et al. [7]. Interior-point methods have been proposed by de Miguel, Friedlander, Nogales and Scholtes [5] and Raghunathan and Biegler [23], while Benson, Shanno, and Vanderbei [2] have p... |

6 |
A new extreme point algorithm and its application in psqp algorithms for solving mathematical programs with linear complementarity constraints
- Zhang, Liu
- 2001
(Show Context)
Citation Context ...traint functions proposed or analyzed by Jiang and Ralph [15] (see [20, Chapter 6] and [16] for local convergence analysis); Tseng and Fukushima [9], who use an ɛ-active set method; and Zhang and Liu =-=[30]-=-, who use an extreme-ray descent method. SQP-based methods for MPECs can be found in Liu et al. [19] and Fletcher et al. [7]. Interior-point methods have been proposed by de Miguel, Friedlander, Nogal... |

4 | Hybrid algorithms with active set identification for mathematical programs with complementarity constraints
- Lin, Fukushima
- 2002
(Show Context)
Citation Context ...ukushima and Pang’s analysis [8] of the smoothing scheme of Facchinei 2set al. [6], and including the penalty approaches analyzed by Hu and Ralph [12] and Huang, Yang, and Zhu [13]. Lin and Fukushima =-=[18]-=- have studied the issue of identifying active constraints in smoothing, regularization, and penalty methods. More recently, Anitescu [1] has studied the “elastic mode” for nonlinear programming, in co... |

3 |
Optimization problems with complementarity constraints: Regularity, optimality conditions, and sensitivity
- Izmailov
- 2004
(Show Context)
Citation Context ...nderbei [2] have performed a computational study involving the LOQO interior-point code and the MacMPEC test set (Leyffer [17]). An anonymous referee has alerted us to a forthcoming paper by Izmailov =-=[14]-=-. We do not have access to an English translation of this paper, but believe that it includes analysis similar to some of that which appears in our proofs below (in particular, the proof of Theorem 3.... |

3 |
Some properties of regularization schemes for MPECs
- Ralph, Wright
- 2003
(Show Context)
Citation Context ... of Theorems 3.2 and 3.3 with proofs that relied on results of Bonnans and Shapiro [3]. The earlier proofs, which are more self-contained but considerably longer, can be found in the technical report =-=[24]-=-. The second author acknowledges the support of the National Science Foundation under Grants #ACI-0082065, #ACI-0113051, #ATM-0086559, and #EIA-0127857. (A.1) Appendix A. Perturbation Results for Regu... |

1 |
An interior-point method for MPECs, LBS working paper
- Miguel, Friedlander, et al.
- 2003
(Show Context)
Citation Context ...reme-ray descent method. SQP-based methods for MPECs can be found in Liu et al. [19] and Fletcher et al. [7]. Interior-point methods have been proposed by de Miguel, Friedlander, Nogales and Scholtes =-=[5]-=- and Raghunathan and Biegler [23], while Benson, Shanno, and Vanderbei [2] have performed a computational study involving the LOQO interior-point code and the MacMPEC test set (Leyffer [17]). An anony... |

1 | Mathematical programs with equilibrium constraints and game theory models in electricity markets - Hu - 2003 |