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35
Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints
 SIAM Journal on Optimization
, 1997
"... Mathematical programs with nonlinear complementarity constraints are reformulated using betterposed 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 ex ..."
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Cited by 35 (0 self)
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Mathematical programs with nonlinear complementarity constraints are reformulated using betterposed 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 lowerlevel 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...
The Global Linear Convergence of a NonInterior PathFollowing Algorithm for Linear Complementarity Problems
 Mathematics of Operations Research
, 1997
"... A noninterior path following algorithm is proposed for the linear complementarity problem. The method employs smoothing techniques introduced by Kanzow. If the LCP is P 0 +R 0 and satisfies a nondegeneracy condition due to Fukushima, Luo, and Pang, then the algorithm is globally linearly converg ..."
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Cited by 31 (3 self)
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A noninterior path following algorithm is proposed for the linear complementarity problem. The method employs smoothing techniques introduced by Kanzow. If the LCP is P 0 +R 0 and satisfies a nondegeneracy condition due to Fukushima, Luo, and Pang, then the algorithm is globally linearly convergent. As with interior point path following methods, the convergence theory relies on the notion of a neighborhood for the central path. However, the choice of neighborhood differs significantly from that which appears in the interior point literature. Numerical experiments are presented that illustrate the significance of the neighborhood concept for this class of methods. 1 Introduction In this paper, we develop a noninterior path following method for the linear complementarity problem: LCP(q;M): Find (x ; y ) 2 IR n \Theta IR n satisfying Mx \Gamma y + q = 0; (1.1) x 0; y 0; (x ) T y = 0; (1.2) where M 2 IR n\Thetan and q 2 IR n . The global line...
QuasiVariational Inequalities, Generalized Nash Equilibria, and MultiLeaderFollower Games JongShi Pang and Masao Fukushima
 Computational Management Science
, 2002
"... The noncooperative multileaderfollower game can be formulated as a generalized Nash equilibrium problem where each player solves a nonconvex mathematical program with equilibrium constraints. Two major deficiencies exist with such a formulation: One is that the resulting Nash equilibrium may no ..."
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Cited by 31 (3 self)
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The noncooperative multileaderfollower game can be formulated as a generalized Nash equilibrium problem where each player solves a nonconvex mathematical program with equilibrium constraints. Two major deficiencies exist with such a formulation: One is that the resulting Nash equilibrium may not exist, due to the nonconvexity in each player's problem; the other is that such a nonconvex Nash game is computationally intractable. In order to obtain a viable formulation that is amenable to practical solution, we introduce a class of remedial models for the multileaderfollower game that can be formulated as generalized Nash games with convexified strategy sets. In turn, a game of the latter kind can be formulated as a quasivariational inequality for whose solution we develop an iterative penalty method.
NonInterior Continuation Methods For Solving Semidefinite Complementarity Problems
 Math. Programming
, 1999
"... There recently has been much interest in noninterior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of blockdiagonal symmetric positive semidefinite real matrices. These ..."
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Cited by 29 (3 self)
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There recently has been much interest in noninterior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of blockdiagonal symmetric positive semidefinite real matrices. These extensions involve the ChenMangasarian class of smoothing functions and the smoothed FischerBurmeister function. Issues such as existence of Newton directions, boundedness of iterates, global convergence, and local superlinear convergence will be studied. Preliminary numerical experience on semidefinite linear programs is also reported. Key words. Semidefinite complementarity problem, smoothing function, noninterior continuation, global convergence, local superlinear convergence. 1 Introduction There recently has been much interest in semidefinite linear programs (SDLP) and, more generally, semidefinite linear complementarity problems (SDLCP), which are extensions of LP and LCP, respecti...
A Global Linear and Local Quadratic Noninterior Continuation Method For Nonlinear Complementarity Problems Based on ChenMangasarian Smoothing Functions
, 1997
"... A noninterior continuation method is proposed for nonlinear complementarity problems. The method improves the noninterior continuation methods recently studied by Burke and Xu [1] and Xu [29]. Our definition of neighborhood for the central path is simpler and more natural. In addition, our continu ..."
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Cited by 22 (2 self)
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A noninterior continuation method is proposed for nonlinear complementarity problems. The method improves the noninterior continuation methods recently studied by Burke and Xu [1] and Xu [29]. Our definition of neighborhood for the central path is simpler and more natural. In addition, our continuation method is based on a broader class of smooth functions introduced by Chen and Mangasarian [7]. The method is shown to be globally linearly and locally quadratically convergent under suitable assumptions. 1 Introduction Let F : R n ! R n be a continuously differentiable function. The nonlinear complementarity problem (NCP) is to find (x; y) 2 R n \Theta R n such that F (x) \Gamma y = 0; (1) x 0; y 0; x T y = 0: (2) Numerous methods have been developed to solve the NCP, for a comprehensive survey see [13, 23]. In this paper, we are interested in developing a noninterior continuation method for the NCP and analyzing its rate of convergence. Department of Management and ...
An implementable activeset algorithm for computing a Bstationary point of a mathematical program with linear complementarity constraints
 SIAM J. Optim
"... Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster poi ..."
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Cited by 22 (4 self)
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Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster points are Bstationary points of the problem. However, the proof has a gap and only shows that each cluster point is an Mstationary point. We discuss this gap and show that Bstationarity can be achieved if the algorithm is modified and an additional error bound condition holds. Key words. MPEC, Bstationary point, ɛactive set, error bound AMS subject classifications. 65K05, 90C30, 90C33
QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints
"... . We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) co ..."
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Cited by 20 (5 self)
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. We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, illconditioning, convexity of the objective, monotonicity and symmetry of the secondlevel problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.maths.mu.OZ.AU/~danny/qpecgendoc.h...
Numerical experience with solving MPECs as NLPs
 Department of Mathematics and Computer Science, University of Dundee, Dundee
, 2002
"... This paper describes numerical experience with solving MPECs as NLPs on a large collection of test problems. The key idea is to use offtheshelf NLP solvers to tackle large instances of MPECs. It is shown that SQP methods are very well suited to solving MPECs and at present outperform Interior Poin ..."
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Cited by 19 (1 self)
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This paper describes numerical experience with solving MPECs as NLPs on a large collection of test problems. The key idea is to use offtheshelf NLP solvers to tackle large instances of MPECs. It is shown that SQP methods are very well suited to solving MPECs and at present outperform Interior Point solvers both in terms of speed and reliability. All NLP solvers also compare very favourably to special MPEC solvers on tests published in the literature.
A NonInterior PredictorCorrector PathFollowing Method for LCP
 Mathematical Programming
, 1997
"... In a previous work the authors introduced a noninterior predictorcorrector path following algorithm for the monotone linear complementarity problem. The method uses ChenHarkerKanzowSmale smoothing techniques to track the central path and employs a refined notion for the neighborhood of the ..."
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Cited by 17 (1 self)
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In a previous work the authors introduced a noninterior predictorcorrector path following algorithm for the monotone linear complementarity problem. The method uses ChenHarkerKanzowSmale smoothing techniques to track the central path and employs a refined notion for the neighborhood of the central path to obtain the boundedness of the iterates under the assumption of monotonicity and the existence of a feasible interior point. With these assumptions, the method is shown to be both globally linearly convergent and locally quadratically convergent. In this paper it is shown that this basic approach is still valid without the monotonicity assumption and regardless of the choice of norm in the definition of the neighborhood of the central path. Furthermore, it is shown that the method can be modified so that only one system of linear equations need to be solved at each iteration without sacrificing either the global or local convergence behavior of the method. The local behavior o...
Complementarity Constraint Qualifications and Simplified BStationarity Conditions for Mathematical Programs with Equilibrium Constraints
, 1998
"... With the aid of some novel complementarity constraint qualifications, we derive some simplied primaldual characterizations of a Bstationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a prog ..."
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Cited by 15 (6 self)
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With the aid of some novel complementarity constraint qualifications, we derive some simplied primaldual characterizations of a Bstationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a program near a feasible point. The simplied results, which rely heavily on a careful dissection and improved understanding of the tangent cone of the feasible region of the program, bypass the combinatorial characterization that is intrinsic to Bstationarity.