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On Solving Mathematical Programs With Complementarity Constraints As Nonlinear Programs
, 2002
"... . We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using classical algorithms and procedures from nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sufficient conditions for their Lagrange multiplier ..."
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Cited by 41 (2 self)
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. We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using classical algorithms and procedures from nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sufficient conditions for their Lagrange multiplier set to be nonempty in two different formulations. MPCCs that have nonempty Lagrange multiplier sets and that satisfy the quadratic growth condition can be approached by the elastic mode with a boundedpenalty parameter. This transformsthe MPCC into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem, which in turn makes it approachable by a sequential quadratic programming algorithm. The robustness of the elastic mode when applied to MPCCs is demonstrated by several numerical examples. 1. Introduction. Complementarity constraints can be used to model numerous economics or mechanics applications [18, 25]....
Modifying SQP for degenerate problems
 Preprint ANL/MCSP6991097, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Ill
, 1997
"... Abstract. Most local convergence analyses of the sequential quadratic programming (SQP) algorithm for nonlinear programming make strong assumptions about the solution, namely, that the active constraint gradients are linearly independent and that there are no weakly active constraints. In this paper ..."
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Cited by 38 (5 self)
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Abstract. Most local convergence analyses of the sequential quadratic programming (SQP) algorithm for nonlinear programming make strong assumptions about the solution, namely, that the active constraint gradients are linearly independent and that there are no weakly active constraints. In this paper, we establish a framework for variants of SQP that retain the characteristic superlinear convergence rate even when these assumptions are relaxed, proving general convergence results and placing some recently proposed SQP variants in this framework. We discuss the reasons for which implementations of SQP often continue to exhibit good local convergence behavior even when the assumptions commonly made in the analysis are violated. Finally, we describe a new algorithm that formalizes and extends standard SQP implementation techniques, and we prove convergence results for this method also. AMS subject classifications. 90C33, 90C30, 49M45 1. Introduction. We
Local behavior of an iterative framework for generalized equations with nonisolated solutions
 MATH. PROGRAM., SER. A
, 2002
"... ..."
An interior point method for mathematical programs with complementarity constraints (MPCCs)
 SIAM JOURNAL ON OPTIMIZATION
, 2003
"... Interior point methods for nonlinear programs (NLP) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primaldual ..."
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Cited by 24 (1 self)
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Interior point methods for nonlinear programs (NLP) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primaldual algorithm has been adapted with a modified step calculation. The algorithm is shown to be superlinearly convergent in the neighborhood of the solution set under assumptions of MPCCLICQ, strong stationarity and upper level strict complementarity. The modification can be easily accommodated within most nonlinear programming interior point algorithms with identical local behavior. Numerical experience is also presented and holds promise for the proposed method.
STABILIZED SEQUENTIAL QUADRATIC PROGRAMMING FOR OPTIMIZATION AND A STABILIZED NEWTONTYPE METHOD FOR VARIATIONAL PROBLEMS WITHOUT CONSTRAINT QUALIFICATIONS
, 2007
"... The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence ..."
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Cited by 24 (14 self)
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The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the secondorder sufficient condition for optimality (SOSC) and the MangasarianFromovitz constraint qualification, or under the strong secondorder sufficient condition for optimality (in that case, without constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming SOSC only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for KarushKuhnTucker systems for variational problems.
An algorithm for degenerate nonlinear programming with rapid local convergence
 SIAM J. Optim
, 2005
"... Abstract. The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. T ..."
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Cited by 22 (0 self)
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Abstract. The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. The first is any conventional method for nonlinear programming that produces estimates of the Lagrange multipliers at each iteration; the second is a technique for estimating the set of active constraint indices; the third is stabilized LagrangeNewton algorithm with rapid local convergence properties. Results concerning rapid local convergence and global convergence of the proposed framework are proved. The approach improves on existing approaches in that less restrictive assumptions are needed for convergence and/or the computational workload at each iteration is lower.
Constraint identification and algorithm stabilization for degenerate nonlinear programs
 Mathematical Programming
, 2003
"... Abstract. In the vicinity of a solution of a nonlinear programming problem at which both strict complementarity and linear independence of the active constraints may fail to hold, we describe a technique for distinguishing weakly active from strongly active constraints. We show that this information ..."
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Cited by 20 (1 self)
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Abstract. In the vicinity of a solution of a nonlinear programming problem at which both strict complementarity and linear independence of the active constraints may fail to hold, we describe a technique for distinguishing weakly active from strongly active constraints. We show that this information can be used to modify the sequential quadratic programming algorithm so that it exhibits superlinear convergence to the solution under assumptions weaker than those made in previous analyses.
On attraction of Newtontype iterates to multipliers violating secondorder sufficiency conditions
, 2009
"... Assuming that the primal part of the sequence generated by a Newtontype (e.g., SQP) method applied to an equalityconstrained problem converges to a solution where the constraints are degenerate, we investigate whether the dual part of the sequence is attracted by those Lagrange multipliers which s ..."
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Cited by 20 (15 self)
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Assuming that the primal part of the sequence generated by a Newtontype (e.g., SQP) method applied to an equalityconstrained problem converges to a solution where the constraints are degenerate, we investigate whether the dual part of the sequence is attracted by those Lagrange multipliers which satisfy secondorder sufficient condition (SOSC) for optimality, or by those multipliers which violate it. This question is relevant at least for two reasons: one is speed of convergence of standard methods; the other is applicability of some recently proposed approaches for handling degenerate constraints. We show that for the class of damped Newton methods, convergence of the dual sequence to multipliers satisfying SOSC is unlikely to occur. We support our findings by numerical experiments. We also suggest a simple auxiliary procedure for computing multiplier estimates, which does not have this
Degenerate Nonlinear Programming with a Quadratic Growth Condition
 Preprint ANL/MCSP7610699, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Ill
"... . We show that the quadratic growth condition and the MangasarianFromovitz constraint qualification imply that local minima of nonlinear programs are isolated stationary points. As a result, when started sufficiently close to such points, an L1 exact penalty sequential quadratic programming algorit ..."
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Cited by 18 (5 self)
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. We show that the quadratic growth condition and the MangasarianFromovitz constraint qualification imply that local minima of nonlinear programs are isolated stationary points. As a result, when started sufficiently close to such points, an L1 exact penalty sequential quadratic programming algorithm will induce at least Rlinear convergence of the iterates to such a local minimum. We construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the MangasarianFromovitz constraint qualification but for which no positive semidefinite augmented Lagrangian exists. We present numerical results obtained using several nonlinear programming packages on this example, and discuss its implications for some algorithms. 1. Introduction. Recently, there has been renewed interest in analyzing and modifying sequential quadratic programming (SQP) algorithms for constrained nonlinear optimization for cases where the traditional regularity cond...