## Basis- and Tripartition Identification for Quadratic Programming and Linear Complementarity Problems - From an interior solution to an optimal basis and viceversa (1996)

Citations: | 3 - 2 self |

### BibTeX

@MISC{Berkelaar96basis-and,

author = {A. B. Berkelaar and B. Jansen and C. Roos and T. Terlaky},

title = {Basis- and Tripartition Identification for Quadratic Programming and Linear Complementarity Problems - From an interior solution to an optimal basis and viceversa },

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal complementary solutions. Maximal complementary solutions can be characterized by optimal (tri)partitions. On the other hand, the solutions provided by simplex--based pivot algorithms are given in terms of complementary bases. A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A tripartition identification algorithm is an algorithm which generates a maximal complementary solution (and its corresponding tripartition), starting from any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal...

### Citations

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(Show Context)
Citation Context ...olution (in the limit). In linear programming the set T is empty, which leads to the existence of strictly complementary solutions. This was first shown by Goldman and Tucker [13] (see e.g. Schrijver =-=[37]-=- and Jansen et al. [19] for an IPM based proof). The resulting partition (B; N ) is called the optimal partition. In this paper we will be concerned with optimal tripartitions. 3.2 Orthogonality Prope... |

683 |
Portfolio Selection: Efficient Diversification of Investments
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Citation Context ... commercial level at all. Berkelaar et al. [4] considered sensitivity analysis using tripartitions instead of bases for QP. As an example they consider the mean-variance portfolio models of Markowitz =-=[30]-=-. These models are often used by financial institutions. ffl Simplex based solvers for both LP and QP are until date more efficient than interior point methods in the case of reoptimization. This is p... |

627 | The Linear Complementarity Problem
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Citation Context ...thods can sometimes suffer from round--off errors with respect to complementarity. The two basic tools that are used in both the BI-algorithm and the TI-algorithm are 1. Principal pivoting (see, e.g. =-=[8, 9, 10, 11, 43, 44]-=-). 2. Elimination based on the orthogonality property of pivot tableaus (see, e.g. [12, 17, 25, 39]). The principal pivot transform was first introduced by Tucker [43, 44]. Cottle and Dantzig were the... |

166 | LOQO: An interior point code for quadratic programming
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Citation Context ... a solution for which x i ? 0 () i 2 B; s i ? 0 () i 2 N ; and it was shown to exist by Guler and Ye [16]. Moreover, they proved that interior point methods such as Carpenter et al. [7] and Vanderbei =-=[46]-=- generate such a solution (in the limit). In linear programming the set T is empty, which leads to the existence of strictly complementary solutions. This was first shown by Goldman and Tucker [13] (s... |

92 |
Bimatrix equilibrium points and mathematical programming
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Citation Context ... solve (LCP) are based on complementary pivot theory ([9, 29]). Lemke [29] describes three different schemes. Lemke's scheme 1, has become known as Lemke's pivot algorithm or Lemke's method (see also =-=[28]-=-). Most of these complementary pivot algorithms go through on so--called almost complementary solutions. Definition 2.8 Let x; s 2 IR n . We call (x; s) an almost complementary solution of the pair LC... |

53 | Linear algebra and its applications,” Harcourt Brace Jovanovish Inc - Strang - 1988 |

48 | On finding primal- and dual-optimal bases
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Citation Context ...lgorithm generates an optimal (tri)partition and a maximal complementary solution, starting from an optimal basis. It is well known that for LP a BI-algorithm exists. This algorithm is due to Megiddo =-=[32]-=- and is strongly polynomial (see also [14, 48]). For linear programming this algorithm is very efficient in practice (see, e.g. [5, 6]), but, to our best knowledge, until now, no such algorithm exists... |

37 |
The simplex method for quadratic programming
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Citation Context ...ceding section, consists of a basic transformation: a set of basic vectors is replaced by its complementary set. Nonnegativity and complementarity conditions are preserved in Lemke's [28] and Wolfe's =-=[47]-=- methods. These algorithms stop if equality conditions are fulfilled. In Keller's [22], CottleDantzig 's [9] and Van de Panne-Whinston's [35] methods equality and complementarity conditions are satisf... |

33 |
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Citation Context ...ed on LP (skew symmetric matrices [19]), QP (bisymmetric matrices) and LCPs with sufficient matrices. Thus, the algorithm is also applicable to any LCP with matrices from the class Psof Kojima et al. =-=[26]-=-. Even more, the algorithm is applicable to LCPs with column sufficient matrices 6 , despite of the fact that no polynomial algorithm to solve such problems exists up till date. However, if we have an... |

30 | A Study and Analysis - Jansen, Spink, et al. |

24 |
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Citation Context ...me practical applications. This can also be said of QP and LCPs. However, there are also situations in which a strictly complementary solution or the knowledge of the optimal partition is needed (see =-=[15]-=-). ffl In commercial OR-packages sensitivity and postoptimal analysis are based on optimal bases. Although several papers have pointed out that this approach can lead to misleading and incorrect concl... |

24 |
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Citation Context ...mplementary pivot algorithms consists of a basic transformation: a set of basic variables is replaced by its complementary set. Many algorithms to solve (LCP) are based on complementary pivot theory (=-=[9, 29]-=-). Lemke [29] describes three different schemes. Lemke's scheme 1, has become known as Lemke's pivot algorithm or Lemke's method (see also [28]). Most of these complementary pivot algorithms go throug... |

18 | Combining interior-point and pivoting algorithms for linear programming
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(Show Context)
Citation Context ...n of an interior point method is a point in the relative interior of the optimal face [16, 21, 31, 33]. Recently the use of combined interior-point and pivot algorithms is being considered (see, e.g. =-=[1]-=-). Both from a theoretical and a practical point of view it is interesting to investigate how maximal complementary solutions and complementary basic solutions can be found from each other. Below some... |

18 |
A convergent criss-cross method
- Terlaky
- 1985
(Show Context)
Citation Context ...at are used in both the BI-algorithm and the TI-algorithm are 1. Principal pivoting (see, e.g. [8, 9, 10, 11, 43, 44]). 2. Elimination based on the orthogonality property of pivot tableaus (see, e.g. =-=[12, 17, 25, 39]-=-). The principal pivot transform was first introduced by Tucker [43, 44]. Cottle and Dantzig were the first to use this transform in their principal pivoting method (see [8, 9, 10, 11]) for linear com... |

17 | A finite criss-cross method for oriented matroids - Terlaky - 1987 |

16 | A computational view of interior-point methods for linear programming
- Gondzio, Terlaky
- 1996
(Show Context)
Citation Context ...on and a maximal complementary solution, starting from an optimal basis. It is well known that for LP a BI-algorithm exists. This algorithm is due to Megiddo [32] and is strongly polynomial (see also =-=[14, 48]-=-). For linear programming this algorithm is very efficient in practice (see, e.g. [5, 6]), but, to our best knowledge, until now, no such algorithm exists for QP and LCPs. In this paper we present a s... |

15 |
Linear and Quadratic Programming in Oriented Matroids
- Todd
- 1985
(Show Context)
Citation Context ... 3. All eigenvalues of M are positive (nonnegative). It is well known [8, 11, 25] that a QP problem can be formulated as an LCP. This goes as follows. Consider the QP in symmetrical form, as found in =-=[25, 35, 42]-=-. min x;z c T x + 1 2 x T C T Cx+ 1 2 z T z s.t. Ax +Bzsb; xs0 This primal QP problem and its dual form yield a pair LCP(q (1) ; M (1) ), where M (1) = 2 4 \GammaR \GammaA A T \GammaQ 3 5 q (1) = 0 @ ... |

13 |
Higher order predictor-corrector interior point methods with application to quadratic objectives
- Carpenter, Lustig, et al.
- 1993
(Show Context)
Citation Context ...solution (x; s) is a solution for which x i ? 0 () i 2 B; s i ? 0 () i 2 N ; and it was shown to exist by Guler and Ye [16]. Moreover, they proved that interior point methods such as Carpenter et al. =-=[7]-=- and Vanderbei [46] generate such a solution (in the limit). In linear programming the set T is empty, which leads to the existence of strictly complementary solutions. This was first shown by Goldman... |

13 | Some Generalizations of the Criss–Cross Method for Quadratic Programming, Mathemathische Operationsforschung und Statistics ser
- Klafszky, Terlaky
- 1992
(Show Context)
Citation Context ...at are used in both the BI-algorithm and the TI-algorithm are 1. Principal pivoting (see, e.g. [8, 9, 10, 11, 43, 44]). 2. Elimination based on the orthogonality property of pivot tableaus (see, e.g. =-=[12, 17, 25, 39]-=-). The principal pivot transform was first introduced by Tucker [43, 44]. Cottle and Dantzig were the first to use this transform in their principal pivoting method (see [8, 9, 10, 11]) for linear com... |

12 |
Duality theory of linear programs - a constructive approach with applications
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(Show Context)
Citation Context ...lgorithm is of both theoretical and practical interest, we will mainly be concerned with the theoretical aspects. For LP there also exists a TI-algorithm. This algorithm is due to Balinski and Tucker =-=[3]-=-. Balinski and Tucker used this algorithm or procedure to proof the existence of strictly complementary solutions for LP by constructing so--called Balinski-Tucker tableaus. In this paper their approa... |

12 |
k-violation linear programming
- Roos, Widmayer
- 1994
(Show Context)
Citation Context ... In linear programming the set T is empty, which leads to the existence of strictly complementary solutions. This was first shown by Goldman and Tucker [13] (see e.g. Schrijver [37] and Jansen et al. =-=[19]-=- for an IPM based proof). The resulting partition (B; N ) is called the optimal partition. In this paper we will be concerned with optimal tripartitions. 3.2 Orthogonality Property of Basic Tableaus W... |

12 |
Pathways to the Optimal Set
- MEGIDDO
- 1989
(Show Context)
Citation Context ...ncide, however non-uniqueness is quite common in practical problems. In case of non-uniqueness, an optimal solution of an interior point method is a point in the relative interior of the optimal face =-=[16, 21, 31, 33]-=-. Recently the use of combined interior-point and pivot algorithms is being considered (see, e.g. [1]). Both from a theoretical and a practical point of view it is interesting to investigate how maxim... |

11 |
Complementary Pivot Theory of
- Cottle, Dantzig
- 1968
(Show Context)
Citation Context ...thods can sometimes suffer from round--off errors with respect to complementarity. The two basic tools that are used in both the BI-algorithm and the TI-algorithm are 1. Principal pivoting (see, e.g. =-=[8, 9, 10, 11, 43, 44]-=-). 2. Elimination based on the orthogonality property of pivot tableaus (see, e.g. [12, 17, 25, 39]). The principal pivot transform was first introduced by Tucker [43, 44]. Cottle and Dantzig were the... |

9 |
Recovering an Optimal LP Basis from an InteriorPoint Solution
- Bixby, Saltzman
- 1991
(Show Context)
Citation Context ...that for LP a BI-algorithm exists. This algorithm is due to Megiddo [32] and is strongly polynomial (see also [14, 48]). For linear programming this algorithm is very efficient in practice (see, e.g. =-=[5, 6]-=-), but, to our best knowledge, until now, no such algorithm exists for QP and LCPs. In this paper we present a strongly polynomial BI-algorithm for QP and LCPs with sufficient matrices (to be defined ... |

9 |
Some Generalizations of the Criss--Cross Method for the Linear Complementarity Problem of Oriented Matroids
- Klafszky, Terlaky
- 1989
(Show Context)
Citation Context ...entarity problems with P -matrices and positive semidefinite matrices (see Section 2). The principal pivot transform is also a basic ingredient of the criss-cross methods presented by Terlaky et al. (=-=[17, 23, 25, 39, 40]-=-). Unless stated explicitly otherwise we will stick to the following notational convention. Lowercase letters will denote scalars or vectors. Matrices will be denoted by capitals, and sets will be den... |

9 | Fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems
- Mitchell
- 1997
(Show Context)
Citation Context ...rticularly the case in mixed integer programming, based on cutting planes and branch and bound techniques. Techniques based on concepts in interior point theory are still in research, see for example =-=[34]-=-. ffl For numerical reasons it sometimes might be necessary to have an optimal basis solution instead of an optimal (tri)partition. Interior point methods can sometimes suffer from round--off errors w... |

8 | A long-step barrier method for convex quadratic programming
- Anstreicher, Hertog, et al.
- 1990
(Show Context)
Citation Context ...ableau, and hence a polynomial TI-algorithm. Proof: Finding a Balinski-Tucker tableau for LCPs consists of solving sub-QP problems or sub-LCPs. Using an interior point method (see, Anstreicher et al. =-=[2]-=- for QP or the algorithm Kojima et al. [26] for LCP) we can solve these subproblems in polynomial time. However, as we know in general we will obtain a maximal complementary solution and not a basic s... |

8 |
Principal pivotal transforms of square matrices
- Tucker
- 1963
(Show Context)
Citation Context ...thods can sometimes suffer from round--off errors with respect to complementarity. The two basic tools that are used in both the BI-algorithm and the TI-algorithm are 1. Principal pivoting (see, e.g. =-=[8, 9, 10, 11, 43, 44]-=-). 2. Elimination based on the orthogonality property of pivot tableaus (see, e.g. [12, 17, 25, 39]). The principal pivot transform was first introduced by Tucker [43, 44]. Cottle and Dantzig were the... |

7 |
The General Quadratic Optimization Problem
- Keller
(Show Context)
Citation Context ...bove transformation defined by the matrix M II is called a principal pivot transformation. This is in fact a block pivot on M II , which is called shortly also as principal pivoting. Following Keller =-=[22]-=- and Van de Panne and Whinston [35] a 1 \Theta 1 principal pivot is called diagonal pivot and a 2 \Theta 2 principal pivot is called exchange pivot. The principal pivot transform as used in so--called... |

7 | Sensitivity analysis in (degenerate) quadratic programming - BERKELAAR, JANSEN, et al. - 1996 |

6 | The role of pivoting in proving some fundamental theorems of linear algebra
- Klafszky, Terlaky
- 1991
(Show Context)
Citation Context ... rowspace of the canonical tableau of �� T and t (k) lies in the column space of the dual canonical tableau. The well--known orthogonality property can now be stated in the following lemma. Lemma =-=3.1 [24]-=- Let B and ~ B denote two bases. Then, t (i) T ~ t (k) = 0; 8i 2 JB ; k 2 J ~ N : Proof: If B = ~ B the proof is obvious. The row space of the canonical tableau spanned by ft (i) : i 2 JB g is the ort... |

6 |
New purification algorithms for linear programming
- KORTANEK, ZHU
- 1988
(Show Context)
Citation Context ...rting basis. The BI-algorithm could also be embedded in combined interior point- pivot algorithms, see e.g. Andersen and Ye [1], or in adapted form can be used for generalized purification procedures =-=[27]-=- and crossover from IPM solvers to Simplex--based solvers. The BI-algorithm has a broad range of applications. The only restriction for a class of matrices is invariance under principal pivot transfor... |

5 |
A General Algorithmic Framework for Quadratic Programming and a Generalization of Edmonds-Fukuda Rule as a Finite Version of Van de Panne-Whinston Method, Mathematical Programming (to appear
- Fukuda, Terlaky
- 1989
(Show Context)
Citation Context ...at are used in both the BI-algorithm and the TI-algorithm are 1. Principal pivoting (see, e.g. [8, 9, 10, 11, 43, 44]). 2. Elimination based on the orthogonality property of pivot tableaus (see, e.g. =-=[12, 17, 25, 39]-=-). The principal pivot transform was first introduced by Tucker [43, 44]. Cottle and Dantzig were the first to use this transform in their principal pivoting method (see [8, 9, 10, 11]) for linear com... |

3 |
Convergence behaviour of some interior point algorithms, Working paper 91-4, The
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(Show Context)
Citation Context ...ncide, however non-uniqueness is quite common in practical problems. In case of non-uniqueness, an optimal solution of an interior point method is a point in the relative interior of the optimal face =-=[16, 21, 31, 33]-=-. Recently the use of combined interior-point and pivot algorithms is being considered (see, e.g. [1]). Both from a theoretical and a practical point of view it is interesting to investigate how maxim... |

3 | Interior-Point Methodology for Linear Programming: Duality, Sensitivity Analysis and Computational Aspects
- Jansen, Roos, et al.
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(Show Context)
Citation Context ...ncide, however non-uniqueness is quite common in practical problems. In case of non-uniqueness, an optimal solution of an interior point method is a point in the relative interior of the optimal face =-=[16, 21, 31, 33]-=-. Recently the use of combined interior-point and pivot algorithms is being considered (see, e.g. [1]). Both from a theoretical and a practical point of view it is interesting to investigate how maxim... |

2 |
The Principal Pivoting Method of Quadratic
- Cottle
- 1967
(Show Context)
Citation Context |

2 |
The symmetric formulation of the simplex method for quadratic programming
- Panne, Whinston
- 1969
(Show Context)
Citation Context ... 3. All eigenvalues of M are positive (nonnegative). It is well known [8, 11, 25] that a QP problem can be formulated as an LCP. This goes as follows. Consider the QP in symmetrical form, as found in =-=[25, 35, 42]-=-. min x;z c T x + 1 2 x T C T Cx+ 1 2 z T z s.t. Ax +Bzsb; xs0 This primal QP problem and its dual form yield a pair LCP(q (1) ; M (1) ), where M (1) = 2 4 \GammaR \GammaA A T \GammaQ 3 5 q (1) = 0 @ ... |

1 |
The Linear Complementarity
- Hertog, Roos, et al.
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Citation Context |

1 | Optimal Bases Versus Optimal Partitions for Postoptimal Analysis in Linear Programming submitted to European - Jansen, Roos, et al. - 1995 |

1 | Limiting Behaviour of the Derivatives of Certain Trajectories Associated with a Monotone Horizontal Linear Complementarity Problem, Working paper - Monteiro, Tsuchiya - 1992 |

1 |
Balinski-Tucker Simplex Tableaus
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- 1995
(Show Context)
Citation Context ...nd connection between optimal bases and optimal tripartitions. Zhang's algorithms [48] are closely related to Megiddo's algorithm for optimal basis identification. Independently, Tijssen and Sierksma =-=[41]-=- obtained similar results. Using the BI-algorithm proposed in this paper and a Balinski-Tucker tableau for LCPs, it is interesting to examine if a similar result as Zhang's, holds for QP and LCPs with... |

1 |
A Combinatorial Equivalence of
- Tucker
- 1960
(Show Context)
Citation Context |

1 |
P Matrices are Just Sufficient, Research Report
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(Show Context)
Citation Context ...s0 implies XM T x = 0 for every vector x. It is called column sufficient (CS) if XMxs0 implies XMx = 0 for every vector x. If it is both row and column sufficient, M is called sufficient (S). Valiaho =-=[45]-=- proved that the class of sufficient matrices is equivalent to the class Psof Kojima et al. [26]. Furthermore, the following inclusions are well known from literature. SS ae BS ae PSD ae S = Psae CS a... |

1 |
On the Strict Complementary Slackness Relation
- Zhang
- 1994
(Show Context)
Citation Context ...on and a maximal complementary solution, starting from an optimal basis. It is well known that for LP a BI-algorithm exists. This algorithm is due to Megiddo [32] and is strongly polynomial (see also =-=[14, 48]-=-). For linear programming this algorithm is very efficient in practice (see, e.g. [5, 6]), but, to our best knowledge, until now, no such algorithm exists for QP and LCPs. In this paper we present a s... |