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18
A survey of the Slemma
 SIAM Review
"... Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as ..."
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Cited by 58 (1 self)
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Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the Slemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. Slemma, Sprocedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities
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 42 (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]....
A timestepping method for stiff multibody dynamics with contact and friction
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 2002
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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.
An algorithm for degenerate nonlinear programming with rapid local convergence
 SIAM J. Optim
, 2005
"... Abstract. The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. T ..."
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Cited by 17 (0 self)
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Abstract. The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. The first is any conventional method for nonlinear programming that produces estimates of the Lagrange multipliers at each iteration; the second is a technique for estimating the set of active constraint indices; the third is stabilized LagrangeNewton algorithm with rapid local convergence properties. Results concerning rapid local convergence and global convergence of the proposed framework are proved. The approach improves on existing approaches in that less restrictive assumptions are needed for convergence and/or the computational workload at each iteration is lower.
Newtontype methods for optimization problems without constraint qualifications
 SIAM Journal on Optimization
, 2004
"... Abstract. 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 singular ..."
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Cited by 16 (12 self)
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Abstract. 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.
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 13 (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...
Nonlinear Programs With Unbounded Lagrange Multiplier Sets
 Preprint ANL/MCSP7960200, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2000
"... . We investigate nonlinear programs that have a nonempty but possibly unbounded Lagrange multiplier set and that satisfy the quadratic growth condition. We show that such programs can be transformed, by relaxing the constraints and adding a linear penalty term to the objective function, into equival ..."
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Cited by 11 (1 self)
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. We investigate nonlinear programs that have a nonempty but possibly unbounded Lagrange multiplier set and that satisfy the quadratic growth condition. We show that such programs can be transformed, by relaxing the constraints and adding a linear penalty term to the objective function, into equivalent nonlinear programs that have differentiable data and a bounded Lagrange multiplier set and that satisfy the quadratic growth condition. As a result we can define, for this type of problem, algorithms that are linearly convergent, using only firstorder information, and superlinearly convergent. 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 conditions do not hold [5, 14, 13, 25, 30]. This research is partly motivated by the fact that largescale nonlinear programming problems tend to be almost degenerate (have large co...
Numerical Behavior of a Stabilized SQP Method for Degenerate NLP Problems
"... In this paper we discuss the application of the stabilized SQP method with constraint identi cation (sSQPa) recently proposed by S. J. Wright [12] for nonlinear programming problems at which strict complementarity and/or linear independence of the gradients of the active constraints may fail to ..."
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Cited by 6 (0 self)
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In this paper we discuss the application of the stabilized SQP method with constraint identi cation (sSQPa) recently proposed by S. J. Wright [12] for nonlinear programming problems at which strict complementarity and/or linear independence of the gradients of the active constraints may fail to hold at the solution. We have collected a number of degenerate problems from dierent sources. Our numerical experiments have shown that the sSQPa is ecient and robust even without the incorporation of a classical globalization technique. One of our goals is therefore to handle NLPs that arise as subproblems in global optimization where degeneracy and infeasibility are important issues. We also discuss and present our work along this direction.
Secondorder negativecurvature methods for boxconstrained and general constrained optimization
, 2009
"... A Nonlinear Programming algorithm that converges to secondorder stationary points is introduced in this paper. The main tool is a secondorder negativecurvature method for boxconstrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is ..."
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Cited by 6 (0 self)
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A Nonlinear Programming algorithm that converges to secondorder stationary points is introduced in this paper. The main tool is a secondorder negativecurvature method for boxconstrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (PowellHestenesRockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to secondorder stationary points in situations in which firstorder methods fail are exhibited.