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LOQO: An interior point code for quadratic programming
, 1994
"... ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex ..."
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Cited by 155 (9 self)
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ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1.
Primaldual interior methods for nonconvex nonlinear programming
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterize ..."
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Cited by 58 (5 self)
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Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penaltybarrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primaldual system similar to that proposed for interior methods. The augmented penaltybarrier function may be interpreted as a merit function for values of the primal and dual variables. An inertiacontrolling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penaltybarrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.
Solving reduced KKT systems in barrier methods for linear and quadratic programming
, 1991
"... In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite. For linear programs, these KKT systems are usually reduced to smaller positivedefinite systems AH −1 A T q = s, where H is a large principal submatrix of ..."
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Cited by 22 (7 self)
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In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite. For linear programs, these KKT systems are usually reduced to smaller positivedefinite systems AH −1 A T q = s, where H is a large principal submatrix of K. These systems can be solved more efficiently, but AH −1 A T is typically more illconditioned than K. In order to improve the numerical properties of barrier implementations, we discuss the use of “reduced KKT systems”, whose dimension and condition lie somewhere in between those of K and AH −1 A T. The approach applies to linear programs and to positive semidefinite quadratic programs whose Hessian H is at least partially diagonal. We have implemented reduced KKT systems in a primaldual algorithm for linear programming, based on the sparse indefinite solver MA27 from the Harwell Subroutine Library. Some features of the algorithm are presented, along with results on the netlib LP test set.
A PrimalDual Algorithm for Minimizing a NonConvex Function Subject to Bound and Linear Equality Constraints
, 1996
"... A new primaldual algorithm is proposed for the minimization of nonconvex objective functions subject to simple bounds and linear equality constraints. The method alternates between a classical primaldual step and a Newtonlike step in order to ensure descent on a suitable merit function. Converge ..."
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Cited by 16 (0 self)
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A new primaldual algorithm is proposed for the minimization of nonconvex objective functions subject to simple bounds and linear equality constraints. The method alternates between a classical primaldual step and a Newtonlike step in order to ensure descent on a suitable merit function. Convergence of a welldefined subsequence of iterates is proved from arbitrary starting points. Algorithmic variants are discussed and preliminary numerical results presented. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA Email : arconn@watson.ibm.com 2 Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU Email : nimg@letterbox.rl.ac.uk 3 Current reports available by anonymous ftp from joyousgard.cc.rl.ac.uk (internet 130.246.9.91) in the directory "pub/reports". 4 Department of Mathematics, Facult'es Universitaires ND de la Paix, 61, rue de Bruxelles, B5000 Namur, Belgium, EU Email : pht@ma...
Iterative solution of augmented systems arising in interior methods
 SIAM JOURNAL ON OPTIMIZATION
, 2007
"... Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization. Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton’s method for a po ..."
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Cited by 9 (1 self)
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Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization. Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton’s method for a point satisfying the perturbed optimality conditions. These equations involve both the primal and dual variables and become increasingly illconditioned as the optimization proceeds. In this context, an iterative linear solver must not only handle the illconditioning but also detect the occurrence of KKT matrices with the wrong matrix inertia. A oneparameter family of equivalent linear equations is formulated that includes the KKT system as a special case. The discussion focuses on a particular system from this family, known as the “doubly augmented system, ” that is positive definite with respect to both the primal and dual variables. This property means that a standard preconditioned conjugategradient method involving both primal and dual variables will either terminate successfully or detect if the KKT matrix has the wrong inertia. Constraint preconditioning is a wellknown technique for preconditioning the conjugategradient method on augmented systems. A family of constraint preconditioners is proposed that provably eliminates the inherent illconditioning in the augmented system. A considerable benefit of combining constraint preconditioning with the doubly augmented system is that the preconditioner need not be applied exactly. Two particular “activese ” constraint preconditioners are formulated that involve only a subset of the rows of the augmented system and thereby may be applied with considerably less work. Finally, some numerical experiments illustrate the numerical performance of the proposed preconditioners and highlight some theoretical properties of the preconditioned matrices.
SUPERVISORY AND EXAMINING COMMITTEE
, 2004
"... Imran Maqsood, candidate for the degree of Doctor of Philosophy, has presented a thesis titled, Development of Simulation and OptimizationBased Decision Support Methodologies for Environmental Systems Management, in an oral examination held on September 3, 2004. The following committee members hav ..."
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Imran Maqsood, candidate for the degree of Doctor of Philosophy, has presented a thesis titled, Development of Simulation and OptimizationBased Decision Support Methodologies for Environmental Systems Management, in an oral examination held on September 3, 2004. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material.
Solving Multistage Stochastic Programs With Tree Dissection
, 1991
"... One component of every multistage stochastic program is a filtration that determines the notion of which random events are observable at each stage of the evolution. Within the context of interiorpoint methods, we describe an efficient preordering technique, called filtered dissection, that takes ..."
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One component of every multistage stochastic program is a filtration that determines the notion of which random events are observable at each stage of the evolution. Within the context of interiorpoint methods, we describe an efficient preordering technique, called filtered dissection, that takes advantage of the filtration's structure to dramatically reduce fillin in the factorization as compared with methods such as the default methods employed by cplexbarrier and loqo. We have implemented this technique as a minor modification to loqo, and it produces a roughly 200fold performance improvement. In particular, we have solved a previouslyunsolvable, realworld, 6stage financial investment problem having 800K equations and 1,200K variables (and 8,192 points in its sample space) using a single processor SGI workstation. The filtered dissection algorithm applies in a natural manner to generic (linear and convex) multistage stochastic programs. The approach promises to eliminate t...
Solving Multistage Stochastic Programs With
, 1991
"... One componentofevery multistage stochastic program is a #ltration that determines the notion of which random events are observable at each stage of the evolution. Within the context of interiorpoint methods, we describe an e#cient preordering technique, called #ltered dissection, that takes a ..."
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One componentofevery multistage stochastic program is a #ltration that determines the notion of which random events are observable at each stage of the evolution. Within the context of interiorpoint methods, we describe an e#cient preordering technique, called #ltered dissection, that takes advantage of the #ltration's structure to dramatically reduce #llin in the factorization as compared with methods such as the default methods employed by cplexbarrier and loqo.Wehave implemented this technique as a minor modi#cation to loqo, and it produces a roughly 200fold performance improvement. In particular, wehave solved a previouslyunsolvable, realworld, 6stage #nancial investment problem having 800K equations and 1,200K variables #and 8,192 points in its sample space# using a single processor SGI workstation.