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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
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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
REGULARIZED SEQUENTIAL QUADRATIC PROGRAMMING METHODS
, 2011
"... We present the formulation and analysis of a new sequential quadratic programming (SQP) method for general nonlinearly constrained optimization. The method pairs a primaldual generalized augmented Lagrangian merit function with a flexible line search to obtain a sequence of improving estimates of t ..."
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Cited by 15 (3 self)
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We present the formulation and analysis of a new sequential quadratic programming (SQP) method for general nonlinearly constrained optimization. The method pairs a primaldual generalized augmented Lagrangian merit function with a flexible line search to obtain a sequence of improving estimates of the solution. This function is a primaldual variant of the augmented Lagrangian proposed by Hestenes and Powell in the early 1970s. A crucial feature of the method is that the QP subproblems are convex, but formed from the exact second derivatives of the original problem. This is in contrast to methods that use a less accurate quasiNewton approximation. Additional benefits of this approach include the following: (i) each QP subproblem is regularized; (ii) the QP subproblem always has a known feasible point; and (iii) a projected gradient method may be used to identify the QP active set when far from the solution.
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.
Properties of the LogBarrier Function on Degenerate Nonlinear Programs
 MATH. OPER. RES
, 1999
"... We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independ ..."
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Cited by 14 (0 self)
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We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independent. When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution, and obtain a semiexplicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution.
INEXACT JOSEPHY–NEWTON FRAMEWORK FOR GENERERALIZED EQUATIONS AND ITS APPLICATIONS TO LOCAL ANALYSIS OF NEWTONIAN METHODS FOR CONSTRAINED OPTIMIZATION ∗
, 2008
"... We propose and analyze a perturbed version of the classical JosephyNewton method for solving generalized equations. This perturbed framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilzed version, sequential quadratically constrained quadratic progr ..."
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Cited by 13 (8 self)
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We propose and analyze a perturbed version of the classical JosephyNewton method for solving generalized equations. This perturbed framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilzed version, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods. For the linearly constrained Lagrangian methods, in particular, we obtain superlinear convergence under the secondorder sufficient optimality condition and the strict Mangasarian–Fromovitz constraint qualification, while previous results in the literature assume (in addition to secondorder sufficiency) the stronger linear independence constraint qualification as well as the strict complementarity condition. For the sequential quadratically constrained quadratic programming methods, we prove primaldual superlinear/quadratic convergence under the same assumptions as above, which also gives a new result.
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.