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Sequential Quadratic Programming
, 1995
"... this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can ..."
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Cited by 115 (2 self)
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this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can
A Sqp Method For General Nonlinear Programs Using Only Equality Constrained Subproblems
 MATHEMATICAL PROGRAMMING
, 1993
"... In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [34] it is much simpler to implement and allows any kind of changes of the working s ..."
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Cited by 48 (2 self)
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In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [34] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQPmethods, as demonstrated by extensive numerical tests.
A reduced Hessian method for largescale constrained optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1995
"... ..."
Methods for nonlinear constraints in optimization calculations
 THE STATE OF THE ART IN NUMERICAL ANALYSIS
, 1996
"... ..."
On the realization of the Wolfe conditions in reduced quasiNewton methods for equality constrained optimization
 SIAM Journal on Optimization
, 1997
"... Abstract. This paper describes a reduced quasiNewton method for solving equality constrained optimization problems. A major difficulty encountered by this type of algorithm is the design of a consistent technique for maintaining the positive definiteness of the matrices approximating the reduced He ..."
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Cited by 5 (0 self)
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Abstract. This paper describes a reduced quasiNewton method for solving equality constrained optimization problems. A major difficulty encountered by this type of algorithm is the design of a consistent technique for maintaining the positive definiteness of the matrices approximating the reduced Hessian of the Lagrangian. A new approach is proposed in this paper. The idea is to search for the next iterate along a piecewise linear path. The path is designed so that some generalized Wolfe conditions can be satisfied. These conditions allow the algorithm to sustain the positive definiteness of the matrices from iteration to iteration by a mechanism that has turned out to be efficient in unconstrained optimization.
A Piecewise LineSearch Technique for Maintaining the Positive Definiteness of the Matrices in the SQP Method
, 1997
"... Abstract. A technique for maintaining the positive definiteness of the matrices in the quasiNewton version of the SQP algorithm is proposed. In our algorithm, matrices approximating the Hessian of the augmented Lagrangian are updated. The positive definiteness of these matrices in the space tangent ..."
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Cited by 4 (2 self)
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Abstract. A technique for maintaining the positive definiteness of the matrices in the quasiNewton version of the SQP algorithm is proposed. In our algorithm, matrices approximating the Hessian of the augmented Lagrangian are updated. The positive definiteness of these matrices in the space tangent to the constraint manifold is ensured by a socalled piecewise linesearch technique, while their positive definiteness in a complementary subspace is obtained by setting the augmentation parameter. In our experiment, the combination of these two ideas leads to a new algorithm that turns out to be more robust and often improves the results obtained with other approaches.
A Reduced Hessian Method for LargeScale Constrained Optimization
 SIAM J. Optimization
, 1993
"... We propose a quasiNewton algorithm for solving large optimization problems with nonlinear equality constraints. It is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations. The algorithm incorporates a correction vector that approximates ..."
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Cited by 1 (0 self)
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We propose a quasiNewton algorithm for solving large optimization problems with nonlinear equality constraints. It is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations. The algorithm incorporates a correction vector that approximates the cross term Z T I/VYp in order to estimate the curvature in both the range and null spaces of the constraints. The algorithm can be considered to be, in some sense, a practical implementation of an algorithm of Coleman and Conn. We give conditions under which local and superlinear convergence is obtained.
SQP and PDIP algorithms for Nonlinear Programming
, 2007
"... Penalty and barrier methods are indirect ways of solving constrained optimization problems, using techniques developed in the unconstrained optimization realm. In what follows we shall give the foundation of two more direct ways of solving constrained optimization problems, namely Sequential Quadrat ..."
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Penalty and barrier methods are indirect ways of solving constrained optimization problems, using techniques developed in the unconstrained optimization realm. In what follows we shall give the foundation of two more direct ways of solving constrained optimization problems, namely Sequential Quadratic Programming (SQP) methods and PrimalDual Interior Point (PDIP) methods. 1 Sequential Quadratic Programming For the derivation of the Sequential Quadratic Programming method we shall use the equality constrained problem minimize f(x) x subject to h(x) = 0, (ECP) where f: R n → R and h: R n → R m are smooth functions. An understanding of this problem is essential in the design of SQP methods for general nonlinear programming problems. The KKT conditions for this problem are given by ∇f(x) + m� λi∇hi(x) = 0 (1a) i=1 1 h(x) = 0 (1b) where λ ∈ R m are the Lagrange multipliers associated with the equality constraints. If we use the Lagrangian L(x, λ) = f(x) + m� λihi(x) (2) we can write the KKT conditions (1) more compactly as ∇x L(x, λ) = 0. (EQKKT) ∇λ L(x, λ) As with Newton’s method unconstrained optimization, the main idead behind SQP is to model problem (ECP) at a given point x (k) by a quadratic programming subrpoblem and then use the solution to this problem to construct a more accurate approximation x (k+1). If we perform a Taylor series expansion of system (EQKKT) about (x (k) , λ (k) ) we obtain ∇x L(x (k) , λ (k))
Leastchange quasiNewton updates for equalityconstrained optimization
, 1999
"... . This paper investigates quasiNewton updates for equalityconstrained optimization in abstract vector spaces. Using a leastchange argument we derive a class of rank3 updates to approximations of the onesided projection of the Hessian of the Lagrangian which keeps the symmetric part positive def ..."
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. This paper investigates quasiNewton updates for equalityconstrained optimization in abstract vector spaces. Using a leastchange argument we derive a class of rank3 updates to approximations of the onesided projection of the Hessian of the Lagrangian which keeps the symmetric part positive definite. By imposing the usual assumptions we are able to prove 1step superlinear convergence for one of these updates. Encouraging numerical results and comparisons with other previously analyzed updates are presented. Key words. quasiNewton update  equalityconstrained optimization  superlinear convergence  variable metric method 1. Introduction and Background QuasiNewton methods for nonlinear optimization problems have been studied extensively since the late 60s. While there are a number of updates and convergence analyses for the unconstrained case (see, e.g.,[10] and [4]), the constrained case has only been discussed more recently, e.g., in [6],[22], [7], [19] and [14], and in Pe...