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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
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.
THE THEORY OF 2REGULARITY FOR MAPPINGS WITH LIPSCHITZIAN DERIVATIVES AND ITS APPLICATIONS TO OPTIMALITY CONDITIONS
, 2002
"... We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2regularity (a certain kind of secondorder regularity) for a once differentiable mapping whose derivative is Lipschitz continuous ..."
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Cited by 17 (14 self)
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We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2regularity (a certain kind of secondorder regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to “righthand side ” perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2regular) equality constraints and optimality conditions for related optimization problems. The class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equationbased reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.
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
Complementarity constraint qualification via the theory of secondorder regularity
 SIAM J. Optim. Forthcoming
, 1999
"... Abstract. We exhibit certain secondorder regularity properties ofparametric complementarity constraints, which are notorious for being irregular in the classical sense. Our approach leads to a constraint qualification in terms of2regularity ofthe mapping corresponding to the subset ofconstraints w ..."
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Cited by 6 (5 self)
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Abstract. We exhibit certain secondorder regularity properties ofparametric complementarity constraints, which are notorious for being irregular in the classical sense. Our approach leads to a constraint qualification in terms of2regularity ofthe mapping corresponding to the subset ofconstraints which must be satisfied as equalities around the given feasible point, while no qualification is required for the rest of the constraints. Under this 2regularity assumption, we derive constructive sufficient conditions for tangent directions to feasible sets defined by complementarity constraints. A special form of primaldual optimality conditions is also obtained. We further show that our 2regularity condition always holds under the piecewise Mangasarian–Fromovitz constraint qualification, but not vice versa. Relations with other constraint qualifications and optimality conditions are also discussed. It is shown that our approach can be useful when alternative ones are not applicable.
CONSTRAINT QUALIFICATIONS
 ENCYCLOPEDIA OF OPERATIONS RESEARCH AND MANAGEMENT SCIENCE
"... We discuss assumptions on the constraint functions that allow constructive description of the geometry of a given set around a given point in terms of the constraints derivatives. Consequences for characterizing solutions of variational and optimization problems are discussed. In the optimization ..."
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Cited by 6 (4 self)
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We discuss assumptions on the constraint functions that allow constructive description of the geometry of a given set around a given point in terms of the constraints derivatives. Consequences for characterizing solutions of variational and optimization problems are discussed. In the optimization case, these include primal and primaldual first and secondorder necessary optimality conditions.
On Optimality Conditions for ConeConstrained Optimization
"... Abstract We consider feasible sets given by conic constraints, where the cone defining the constraints is convex with nonempty interior. We study the case where the feasible set is not assumed to be regular in the classical sense of Robinson and obtain a constructive description of the tangent cone ..."
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Abstract We consider feasible sets given by conic constraints, where the cone defining the constraints is convex with nonempty interior. We study the case where the feasible set is not assumed to be regular in the classical sense of Robinson and obtain a constructive description of the tangent cone under a certain new secondorder regularity condition. This condition contains classical regularity as a special case, while being weaker when constraints are twice differentiable. Assuming that the cone defining the constraints is finitely generated, we also derive a special form of primaldual optimality conditions for the corresponding constrained optimization problem. Our results subsume optimality conditions for both the classical regular and secondorder regular cases, while still being meaningful in the more general setting in the sense that the multiplier associated with the objective function is nonzero.
Mathematical Programming manuscript No. (will be inserted by the editor)
, 2006
"... On attraction of Newtontype iterates to multipliers violating secondorder sufficiency conditions Dedicated to Professor Stephen Robinson on the occasion of his 65th birthday. The second author remembers, with a sense of privilege, the courses and advice he received from Professor Robinson during h ..."
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On attraction of Newtontype iterates to multipliers violating secondorder sufficiency conditions Dedicated to Professor Stephen Robinson on the occasion of his 65th birthday. The second author remembers, with a sense of privilege, the courses and advice he received from Professor Robinson during his stay at UWMadison.
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"... Secondorder necessary optimality conditions for problems without a priori normality assumptions ..."
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Secondorder necessary optimality conditions for problems without a priori normality assumptions