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14
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.
On attraction of linearly constrained Lagrangian methods and of stabilized and quasiNewton SQP methods to critical multipliers
 MATHEMATICAL PROGRAMMING
, 2009
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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
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.
ON LOCAL CONVERGENCE OF SEQUENTIAL QUADRATICALLYCONSTRAINED QUADRATICPROGRAMMING TYPE METHODS, WITH AN EXTENSION TO VARIATIONAL PROBLEMS ∗
, 2005
"... We consider the class of quadraticallyconstrained quadraticprogramming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (2002) showed superlinear convergence of the primal sequence under the MangasarianFromovi ..."
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Cited by 10 (4 self)
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We consider the class of quadraticallyconstrained quadraticprogramming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (2002) showed superlinear convergence of the primal sequence under the MangasarianFromovitz constraint qualification and the quadratic growth condition. Quadratic convergence of the primaldual sequence was established by Fukushima, Luo and Tseng (2003) under the assumption of convexity, the Slater constraint qualification, and a strong secondorder sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primaldual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a secondorder sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a DennisMoré type condition. Key words. Quadratically constrained quadratic programming, KarushKuhnTucker system, variational inequality, quadratic convergence, superlinear convergence, DennisMoré condition.
A NOTE ON UPPER LIPSCHITZ STABILITY, ERROR BOUNDS, AND CRITICAL MULTIPLIERS FOR LIPSCHITZCONTINUOUS KKT SYSTEMS
, 2012
"... We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qual ..."
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Cited by 8 (5 self)
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We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than secondorder sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.
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.
Computable primal error bounds based on the augmented Lagrangian and Lagrangian relaxation algorithms
, 2006
"... For a given iterate generated by the augmented Lagrangian or the Lagrangian relaxation based method, we derive estimates for the distance to the primal solution of the underlying optimization problem. The estimates are obtained using some recent contributions to the sensitivity theory, under appropr ..."
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Cited by 1 (1 self)
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For a given iterate generated by the augmented Lagrangian or the Lagrangian relaxation based method, we derive estimates for the distance to the primal solution of the underlying optimization problem. The estimates are obtained using some recent contributions to the sensitivity theory, under appropriate first or second order sufficient optimality conditions. The given estimates hold in situations where known (algorithmindependent) error bounds may not apply. Examples are provided which show that the estimates are sharp.
Copyright © 2005 SBMAC
"... www.scielo.br/cam Numerical results for a globalized activeset Newton method for mixed complementarity problems ..."
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www.scielo.br/cam Numerical results for a globalized activeset Newton method for mixed complementarity problems