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17
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.
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
KarushKuhnTucker systems: regularity conditions, error bounds and a class of Newtontype methods
 MATH. PROGRAM., SER. A
, 2003
"... We consider optimality systems of KarushKuhnTucker (KKT) type, which arise, for example, as primaldual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newtontype methods for such systems. An exhaustive compariso ..."
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Cited by 14 (12 self)
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We consider optimality systems of KarushKuhnTucker (KKT) type, which arise, for example, as primaldual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newtontype methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems, such as quasiregularity or semistability (equivalently, the R0property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local algorithms. We propose a family of local Newtontype algorithms. This family contains some known activeset Newton methods, as well as some new methods. Regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth Newton methods and sequential quadratic programming methods.
A class of activeset Newton methods for mixed complementarity problems
 SIAM J. OPTIM
, 2004
"... Based on the identification of indices active at a solution of the mixed complementarity problem (MCP), we propose a class of Newton methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedur ..."
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Cited by 12 (8 self)
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Based on the identification of indices active at a solution of the mixed complementarity problem (MCP), we propose a class of Newton methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedure and the nondegeneracy condition needed for the convergence of the resulting Newton method are individually and collectively strictly weaker than the property of semistability of a solution. Thus the local superlinear convergence conditions of the presented method are weaker than conditions required for the semismooth (generalized) Newton methods applied to MCP reformulations. Moreover, they are also weaker than convergence conditions of the linearization (Josephy–Newton) method. For the special case of optimality systems with primaldual structure, we further consider the question of superlinear convergence of primal variables. We illustrate our theoretical results with numerical experiments on some specially constructed MCPs whose solutions do not satisfy the usual regularity assumptions.
Optimality conditions for irregular inequalityconstrained problems
 SIAM J. Control Optim
"... 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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Cited by 9 (8 self)
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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.
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.
Superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems
 SIAM J. Optim
"... Abstract. We propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular. Compared to other techniques for computing singular solutions, a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in t ..."
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Cited by 5 (3 self)
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Abstract. We propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular. Compared to other techniques for computing singular solutions, a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis. Important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equationbased reformulations of the nonlinear complementarity problems. Reformulations of complementarity problems serve both as illustration of and motivation for our approach, and one of them we consider in detail. We show that the proposed method possesses local superlinear/quadratic convergence under reasonable assumptions. We further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent Newtontype algorithms for solving complementarity problems, which are typically based on nonsmooth reformulations. Therefore our approach appears to be an interesting complement to the existing ones.
Numerical results for a globalized activeset Newton method for mixed complementarity problems
 COMPUTATIONAL AND APPLIED MATHEMATICS
, 2004
"... We discuss a globalization scheme for a class of activeset Newton methods for solving the mixed complementarity problem (MCP), which was proposed by the authors in [3]. The attractive features of the local phase of the method are that it requires solving only one system of linear equations per iter ..."
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Cited by 4 (4 self)
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We discuss a globalization scheme for a class of activeset Newton methods for solving the mixed complementarity problem (MCP), which was proposed by the authors in [3]. The attractive features of the local phase of the method are that it requires solving only one system of linear equations per iteration, yet the local superlinear convergence is guaranteed under extremely mild assumptions, in particular weaker than the property of semistability of an MCP solution. Thus the local superlinear convergence conditions of the method are weaker than conditions required for the semismooth (generalized) Newton methods and also weaker than convergence conditions of the linearization (Josephy–Newton) method. Numerical experiments on some test problems are presented, including results on the MCPLIB collection for the globalized version.
THE JOSEPHY–NEWTON METHOD FOR SEMISMOOTH GENERALIZED EQUATIONS AND SEMISMOOTH SQP FOR OPTIMIZATION
, 2011
"... While generalized equations with differentiable singlevalued base mappings and the associated Josephy–Newton method have been studied extensively, the setting with semismooth base mapping had not been previously considered (apart from the two special cases of usual nonlinear equations and of Karush ..."
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Cited by 4 (4 self)
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While generalized equations with differentiable singlevalued base mappings and the associated Josephy–Newton method have been studied extensively, the setting with semismooth base mapping had not been previously considered (apart from the two special cases of usual nonlinear equations and of KarushKuhnTucker optimality systems). We introduce for the general semismooth case appropriate notions of solution regularity and prove local convergence of the corresponding Josephy–Newton method. As an application, we immediately recover the known primaldual local convergence properties of semismooth SQP, but also obtain some new results that complete the analysis of the SQP primal rate of convergence, including its quasiNewton variant. Key words: generalized equation, Bdifferential, generalized Jacobian, BDregularity, CDregularity,