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38
Local convergence of SQP methods for Mathematical Programs with Equilibrium Constraints
, 2002
"... Recently, it has been shown that Nonlinear Programming solvers can successfully solve a range of Mathematical Programs with Equilibrium Constraints (MPECs). ..."
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Cited by 72 (19 self)
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Recently, it has been shown that Nonlinear Programming solvers can successfully solve a range of Mathematical Programs with Equilibrium Constraints (MPECs).
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]....
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.
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...
A Superlinearly Convergent Sequential Quadratically Constrained Quadratic Programming Algorithm For Degenerate Nonlinear Programming
 SIAM Journal on Optimization
"... . We present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the MangasarianFromovitz constraint qualification and the quadratic growth condition. This convergence result is obtained despite the potential lack of a locally convex augmented Lagrangian. The algori ..."
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Cited by 18 (2 self)
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. We present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the MangasarianFromovitz constraint qualification and the quadratic growth condition. This convergence result is obtained despite the potential lack of a locally convex augmented Lagrangian. The algorithm solves a succession of subproblems that have quadratic objective and quadratic constraints, both possibly nonconvex. By the use of a trustregion constraint we guarantee that any stationary point of the subproblem induces superlinear convergence which avoids the problem of computing a global minimum. 1. Introduction. Recently, there has been renewed interest in analyzing and modifying the algorithms for constrained nonlinear optimization for cases where the traditional regularity conditions do not hold [5, 12, 11, 20, 24, 23]. This research has been motivated by the fact that largescale nonlinear programming problems tend to be almost degenerate (have large condition numbers for the Jac...
NEWTONTYPE METHODS FOR OPTIMIZATION PROBLEMS WITHOUT CONSTRAINT QUALIFICATIONS
 SIAM J. OPTIMIZATION
, 2004
"... We consider equalityconstrained optimization problems, where a given solution may not satisfy any constraint qualification, but satisfies the standard secondorder sufficient condition for optimality. Based on local identification of the rank of the constraints degeneracy via the singularvalue d ..."
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Cited by 17 (13 self)
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We consider equalityconstrained optimization problems, where a given solution may not satisfy any constraint qualification, but satisfies the standard secondorder sufficient condition for optimality. Based on local identification of the rank of the constraints degeneracy via the singularvalue decomposition, we derive a modified primaldual optimality system whose solution is locally unique, nondegenerate, and thus can be found by standard Newtontype techniques. Using identification of active constraints, we further extend our approach to mixed equality and inequalityconstrained problems, and to mathematical programs with complementarity constraints (MPCC). In particular, for MPCC we obtain a local algorithm with quadratic convergence under the secondorder sufficient condition only, without any constraint qualifications, not even the special MPCC constraint qualifications.
On attraction of linearly constrained Lagrangian methods and of stabilized and quasiNewton SQP methods to critical multipliers
 MATHEMATICAL PROGRAMMING
, 2009
"... ..."
Examples of dual behaviour of Newtontype methods on optimization problems with degenerate constraints
 Computational Optimization and Applications
"... discuss possible scenarios of behaviour of the dual part of sequences generated by primaldual Newtontype methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a so ..."
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Cited by 16 (10 self)
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discuss possible scenarios of behaviour of the dual part of sequences generated by primaldual Newtontype methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a socalled critical multiplier (which, in particular, violates some secondorder sufficient conditions for optimality), the latter appearing to be a typical scenario when critical multipliers exist; (c) convergence to a noncritical multiplier. The case of mathematical programs with complementarity constraints is also discussed. We illustrate those scenarios with examples, and discuss consequences for the speed of convergence. We also put together a collection of examples of optimization problems with constraints violating some standard constraint qualifications, intended for preliminary testing of existing algorithms on degenerate problems, or for developing special new algorithms designed to deal with constraints degeneracy. Keywords Degenerate constraints · Secondorder sufficiency · Newton method · SQP