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Inertiacontrolling methods for general quadratic programming
 SIAM Review
, 1991
"... Abstract. Activeset quadratic programming (QP) methods use a working set to define the search direction and multiplier estimates. In the method proposed by Fletcher in 1971, and in several subsequent mathematically equivalent methods, the working set is chosen to control the inertia of the reduced ..."
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Cited by 34 (3 self)
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Abstract. Activeset quadratic programming (QP) methods use a working set to define the search direction and multiplier estimates. In the method proposed by Fletcher in 1971, and in several subsequent mathematically equivalent methods, the working set is chosen to control the inertia of the reduced Hessian, which is never permitted to have more than one nonpositive eigenvalue. (We call such methods inertiacontrolling.) This paper presents an overview of a generic inertiacontrolling QP method, including the equations satisfied by the search direction when the reduced Hessian is positive definite, singular and indefinite. Recurrence relations are derived that define the search direction and Lagrange multiplier vector through equations related to the KarushKuhnTucker system. We also discuss connections with inertiacontrolling methods that maintain an explicit factorization of the reduced Hessian matrix. Key words. Nonconvex quadratic programming, activeset methods, Schur complement, Karush
A Schurcomplement method for sparse quadratic programming
 UNIV CA SYSTEMS OPTIMIZATION LAB
, 1987
"... In applying activeset methods to sparse quadratic programs, it is desirable to utilize existing sparsematrix techniques. We describe a quadratic programming method based on the classical Schur complement. Its key feature is that much of the linear algebraic work associated with an entire sequence ..."
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Cited by 16 (3 self)
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In applying activeset methods to sparse quadratic programs, it is desirable to utilize existing sparsematrix techniques. We describe a quadratic programming method based on the classical Schur complement. Its key feature is that much of the linear algebraic work associated with an entire sequence of iterations involves a fixed sparse factorization. Updates are performed at every iteration to the factorization of a smaller matrix, which may be treated as dense or sparse. The use of a fixed sparse factorization allows an “offthe shelf ” sparse equation solver to be used repeatedly. This feature is ideally suited to problems with structure that can be exploited by a specialized factorization. Moreover, improvements in efficiency derived from exploiting new parallel and vector computer architectures are immediately applicable. An obvious application of the method is in sequential quadratic programming methods for nonlinearly constrained optimization, which require solution of a sequence of closely related quadratic programming subproblems. We discuss some ways in which the known relationship between successive problems can be exploited.
Some Generalizations Of The CrissCross Method For Quadratic Programming
 MATH. OPER. UND STAT. SER. OPTIMIZATION
, 1992
"... Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. ..."
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Cited by 13 (8 self)
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Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite crisscross method, based on leastindex resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used. In the last section some special cases and two further variants of the quadratic crisscross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty's `Bard type schema' in the P matrix case.
A Cholesky dual method for proximal piecewise linear programming
, 1994
"... this paper we present another highly specialized method for the solution of problem (1.1). Our dual method solves (1.3) with a classical activeset strategy [Fle87, GMW81]. It provides a lowstorage implementation of the algorithm in [Kiw89] by replacing its QR factorization with the Cholesky factor ..."
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Cited by 6 (1 self)
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this paper we present another highly specialized method for the solution of problem (1.1). Our dual method solves (1.3) with a classical activeset strategy [Fle87, GMW81]. It provides a lowstorage implementation of the algorithm in [Kiw89] by replacing its QR factorization with the Cholesky factorization of active general constraints. The workspace is reduced from about 3n
Ecole Doctorale: MATISSE présentée par
, 2013
"... Je tiens tout d’abord à remercier mes encadrants. Après avoir grandemment contribué à me faire venir enseigner à l’Université, Eric MatznerLøber m’a fait confiance pour ce travail de recherche; je tiens à lui témoigner toute ma reconnaissance et mes remerciements à son égard dépassent très largemen ..."
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Je tiens tout d’abord à remercier mes encadrants. Après avoir grandemment contribué à me faire venir enseigner à l’Université, Eric MatznerLøber m’a fait confiance pour ce travail de recherche; je tiens à lui témoigner toute ma reconnaissance et mes remerciements à son égard dépassent très largement le cadre professionnel. Arnaud Guyader a accepté sans hésitation de m’aider au moment même où les difficultés à venir s’annonçaient grandes. Je le remercie infiniment pour son écoute et sa très grande disponibilité; ce manuscrit lui doit beaucoup. Enfin, rendons à César ce qui appartient à César: l’idée de la méthode présentée ici est tout droit sortie de l’esprit prodigieux de Nicolas Hengartner. En m’invitant deux fois à Los Alamos ces dernières années, il m’a donné le privilège de travailler à ses côtés. Je suis désormais au moins sûr d’une chose: le génie et l’enthousiasme sont liés! Je tiens à remercier chaleureusement Cécile Durot et Sylvain Sardy pour l’intérêt qu’ils ont porté à mon travail en acceptant de rapporter sur cette thèse. Je suis également très reconnaissant à Christophe Abraham et Gérard Biau d’avoir bien voulu prendre sur leur temps précieux pour faire partie du jury. Je voudrais aussi remercier Marie de Tayrac pour m’avoir fourni les données médicales permettant
Edmonds Fukuda Rule And A General Recursion For Quadratic Programming
"... A general framework of nite algorithms is presented here for quadratic programming. This algorithm is a direct generalization of Van der Heyden's algorithm for the linear complementarity problem and Jensen's `relaxed recursive algorithm', which was proposed for solution of Oriented Ma ..."
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A general framework of nite algorithms is presented here for quadratic programming. This algorithm is a direct generalization of Van der Heyden's algorithm for the linear complementarity problem and Jensen's `relaxed recursive algorithm', which was proposed for solution of Oriented Matroid programming problems. The validity of this algorithm is proved the same way as the finiteness of the crisscross method is proved. The second part of this paper contains a generalization of EdmondsFukuda pivoting rule for quadratic programming. This generalization can be considered as a finite version of Van de Panne  Whinston algorithm and so it is a simplex method for quadratic programming. These algorithms uses general combinatorial type ideas, so the same methods can be applied for oriented matroids as well. The generalization of these methods for oriented matroids is a subject of another paper.