## Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints (1997)

Venue: | SIAM Journal on Optimization |

Citations: | 36 - 0 self |

### BibTeX

@ARTICLE{Jiang97smoothsqp,

author = {Houyuan Jiang and Daniel Ralph},

title = {Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints},

journal = {SIAM Journal on Optimization},

year = {1997},

volume = {10},

pages = {779--808}

}

### Years of Citing Articles

### OpenURL

### Abstract

Mathematical programs with nonlinear complementarity constraints are reformulated using better-posed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations. We present two globally convergent algorithms based on sequential quadratic programming, SQP, as applied in exact penalty methods for nonlinear programs. Global convergence of the implicit smooth SQP method depends on existence of a lower-level nondegenerate (strictly complementary) limit point of the iteration sequence. Global convergence of the explicit smooth SQP method depends on a weaker property, i.e. existence of a limit point at which a generalized constraint qualification holds. We also discuss some practical matter...

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Citation Context ...h can prevent superlinear convergence of an SQP method that uses an exact penalty merit function unless a second-order correction to the feasibility of the iterate is performed at each iteration. See =-=[11, 36]-=-. In order to study the rate of convergence of implicit smooth SQP, further conditions such as the LICQ, the second order sufficient condition, careful update rules of the matrix sequence fW k g, etc.... |

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Citation Context ...ightly stronger than those given in [16] to keep notation simple. When (5) is defined by smooth (continuously differentiable) functions, the GLICQ and GMFCQ reduce to the classical LICQ and MFCQ, see =-=[28, 31]-=-; and, as in the smooth case, the GLICQ implies the GMFCQ. However the CRCQ usually used in the smooth case [17] is stronger than the smooth version of the GCRCQ in that the former requires constant r... |

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Citation Context ...imation problems is improved. This opens the way to use sequential quadratic programming (SQP) methods from classical nonlinear programming. The methods presented in this paper follow some ideas from =-=[8, 12]-=- which try to use well-developed numerical methods for the solution of smooth nonlinear programs. In [8], smooth nonlinear programs of the type (4) are formed and assumed to be solvable by an unspecif... |

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Citation Context ...osing an SQP method in [12]. Corresponding to / is the function OE(a; b) = p a 2 + b 2 \Gamma (a + b), which is now known as the Fischer-Burmeister function [9]. The introduction of / originates from =-=[21]-=- for handling linear complementarity problems. If (a; b; c) 6= (0; 0; 0), then / is smooth at (a; b; c) with r/(a; b; c) = (p; q; r) such that p = a p a 2 + b 2 + c 2 \Gamma 1; q = b p a 2 + b 2 + c 2... |

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Citation Context ... the modified QP (18) is replaced by the QP (16) to generate the search direction in the above algorithm, then our SQP method is very similar to classical SQP methods for smooth nonlinear programming =-=[15, 36]. Th-=-e difference is that here we anticipate nonsmoothness of /. If further �� is treated as a parameter rather than a variable, namely the last equation in (16) is omitted at each iteration, then the ... |

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Citation Context ...\Gamma (1\Gamma) 4 ( p a 2 + c 2 + a)( p b 2 + c 2 + b) OE(a; b) = [ p a 2 + b 2 \Gamma (a + b)] \Gamma (1 \Gamma ) maxfa; 0g maxfb; 0g wheres2 (0; 1] is a parameter. The function OE is introduced in =-=[3]-=- for solving nonlinear complementarity problems. Whens= 1, OE reduces to the Fischer-Burmeister function in Example 7.1. If c 6= 0, then / is smooth at (a; b; c) and r/(a; b; c) = (p; q; r) with p = (... |

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Citation Context ...imation problems is improved. This opens the way to use sequential quadratic programming (SQP) methods from classical nonlinear programming. The methods presented in this paper follow some ideas from =-=[8, 12]-=- which try to use well-developed numerical methods for the solution of smooth nonlinear programs. In [8], smooth nonlinear programs of the type (4) are formed and assumed to be solvable by an unspecif... |

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Citation Context ...the MPEC (1) is equivalent to the smooth nonlinear program (NLP) obtained by writing the complementarity condition F (x; y) ? y as an inner product F (x; y) T y = 0. Unfortunately, it has been proved =-=[4]-=- that the Mangasarian-Fromovitz Constraint Qualification does not hold at any feasible point of this smooth NLP even if the usual inequality constraints g(x; y)s0 are omitted and the lower-level NCP p... |

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Citation Context ...timality conditions in the following proposition. These optimality conditions also hold under the next assumption: Piecewise Affine Constraint Condition (PACC): Both g and h are piecewise affine. See =-=[41]-=- for a proof of generalized stationarity under the PACC. Proposition 2.1 Suppose u is a local minimizer of the nonsmooth program (5) and one of the GCRCQ, GLICQ, GMFCQ or PACC holds at u . Then u is a... |

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Citation Context ...er, a numerical investigation will be pursued in a future publication. We mention that the development of numerical methods for the solution of MPEC is at a less advanced stage than optimality theory =-=[4, 26, 27, 28, 29, 30, 32, 33, 34, 40, 43]-=-. When the upper-level constraints exclude y, i.e. take the form g(x)s0, the implicit function approach may be possible. In this approach it is assumed that y can be found as a function of x by solvin... |

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Citation Context ... 0]; r 2 [\Gamma1; 1]g: Example 7.3 /(a; b; c) = p a 2 + b 2 + ab + c 2 \Gamma (a + b) OE(a; b) = p a 2 + b 2 + ab \Gamma (a + b) wheres2 [\Gamma2; 2) is a parameter. The function OE is introduced in =-=[22]-=- for solving nonlinear complementarity problems. Apparently, whens= 0, OE reduces to the FischerBurmeister function (Example 7.1), and whens= \Gamma2, OE reduces to the min function (Example 7.2). So ... |

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Citation Context ...er, a numerical investigation will be pursued in a future publication. We mention that the development of numerical methods for the solution of MPEC is at a less advanced stage than optimality theory =-=[4, 26, 27, 28, 29, 30, 32, 33, 34, 40, 43]-=-. When the upper-level constraints exclude y, i.e. take the form g(x)s0, the implicit function approach may be possible. In this approach it is assumed that y can be found as a function of x by solvin... |

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Citation Context ...er, a numerical investigation will be pursued in a future publication. We mention that the development of numerical methods for the solution of MPEC is at a less advanced stage than optimality theory =-=[4, 26, 27, 28, 29, 30, 32, 33, 34, 40, 43]-=-. When the upper-level constraints exclude y, i.e. take the form g(x)s0, the implicit function approach may be possible. In this approach it is assumed that y can be found as a function of x by solvin... |

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Citation Context ...gests that well-developed nonlinear programming theory and numerical methods are not readily applicable for solving this form of MPEC: the feasible set of the smooth NLP is numerically ill posed. See =-=[19, 28]-=- for more discussions and numerical examples. Instead we let w = F (x; y) and substitute a nonsmooth equation \Phi(y; w) = 0 2 ! m , constructed using the Fischer-Burmeister functional [9] for example... |

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Citation Context ...mma (a + b): This function is used for proposing an SQP method in [12]. Corresponding to / is the function OE(a; b) = p a 2 + b 2 \Gamma (a + b), which is now known as the Fischer-Burmeister function =-=[9]-=-. The introduction of / originates from [21] for handling linear complementarity problems. If (a; b; c) 6= (0; 0; 0), then / is smooth at (a; b; c) with r/(a; b; c) = (p; q; r) such that p = a p a 2 +... |

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Citation Context ...entiable) functions, the GLICQ and GMFCQ reduce to the classical LICQ and MFCQ, see [28, 31]; and, as in the smooth case, the GLICQ implies the GMFCQ. However the CRCQ usually used in the smooth case =-=[17]-=- is stronger than the smooth version of the GCRCQ in that the former requires constant rank of submatrices of rows of the Jacobian G 0 (u) for u near u . (We mention an example to distinguish these tw... |

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Citation Context ... to the modified QP (18)). This fact can be computationally significant as n is often much smaller than m. The feasibility of the QP (16) is a serious issue in the context of MPEC. Fukushima and Pang =-=[13]-=- discussed it from a different angle, namely for mathematical programs with linear complementarity constraints. We remark that the P 0 property assumed in our paper is not necessarily required in [13]... |

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Citation Context ... include the penalty interior-point algorithm (PIPA for short) [28], smoothing methods [8, 12], which are related to the interior-point approach, and piecewise sequential quadratic programming (PSQP) =-=[28, 29, 37]-=-. Apart from this paper, the only implementations of these algorithms we know of that handle joint upper-level constraints are discussed in [19]. Penalty interior-point algorithms converge globally un... |

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Citation Context ...QP methods play a critical role, in particular SQP fails if one of the associated quadratic programs is infeasible. In order to overcome QP infeasibility, some modifications have been introduced; see =-=[1, 2]. Ou-=-r strategy below is similar to that proposed in [1, 2] but with several notable differences. A modified quadratic program of (16) is defined as follows: min d2! n+2m+1 ;��2! l rf(x; y) T (dx; dy) ... |

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Citation Context ... problem like min x;y;w;�� f(x; y) subject to g(x; y)s0 F (x; y) \Gamma w = 0 \Psi(y; w; ��) = 0; e �� \Gamma 1 = 0; (3) where e is Euler's constant, so that the last constraint requires �=-=��� = 0 (c.f. [18] for complem-=-entarity problems); or by approximately solving the following problem for a sequence of values �� = �� k ! 0, min x;y;w f(x; y) subject to g(x; y)s0 F (x; y) \Gamma w = 0 \Psi(y; w; ��) = ... |

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Citation Context ... to g(x)s0. This nonsmooth problem can be tackled by bundle methods as proposed in [23, 24, 33, 34] or using another nonsmooth optimization method such as Shor's R-algorithm as implemented in SolvOpt =-=[25]-=-; see [7] for some computational comparisons. However with mixed upper-level constraints, i.e. involving y and possibly x, the implicit programming approach transforms an MPEC into a problem with nond... |

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Citation Context ...s, and the MPEC is collapsed to the problem of minimizing the nondifferentiable objective function f(x; y(x)) subject to g(x)s0. This nonsmooth problem can be tackled by bundle methods as proposed in =-=[23, 24, 33, 34]-=- or using another nonsmooth optimization method such as Shor's R-algorithm as implemented in SolvOpt [25]; see [7] for some computational comparisons. However with mixed upper-level constraints, i.e. ... |

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Citation Context ...s, and the MPEC is collapsed to the problem of minimizing the nondifferentiable objective function f(x; y(x)) subject to g(x)s0. This nonsmooth problem can be tackled by bundle methods as proposed in =-=[23, 24, 33, 34]-=- or using another nonsmooth optimization method such as Shor's R-algorithm as implemented in SolvOpt [25]; see [7] for some computational comparisons. However with mixed upper-level constraints, i.e. ... |

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Citation Context ...each fixed x. We omit equality constraints in the upper-level for simplicity, but these can easily be handled and would be useful for the following case. Lower-level mixed complementarity constraints =-=[7]-=- can be dealt with quite easily by moving equations and their associated variables to the upper level. For example, consider the following lower-level mixed complementarity constraints F 1 (x; y; z) =... |

6 |
Refinements of necessary optimality conditions in nondifferentiable programming. II. Optimality and stability in mathematical programming
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Citation Context ...od to be presented in Section 6 --- it is shown that limit points of the sequence of approximate solutions of the parametric nonlinear programs satisfy generalized Karush-Kuhn-Tucker (KKT) conditions =-=[16]-=- given in terms of the Clarke generalized derivatives [5]. We call this an explicit smoothing method because the smoothing parameter is updated separately from the direction-finding process. In [12] a... |

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