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Complementarity Modeling of Hybrid Systems
, 1998
"... A complementarity framework is described for the modeling of certain classes of mixed continuous /discrete dynamical systems. The use of such a framework is wellknown for mechanical systems with inequality constraints, but we give a more general formulation which applies for instance also to sys ..."
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Cited by 51 (11 self)
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A complementarity framework is described for the modeling of certain classes of mixed continuous /discrete dynamical systems. The use of such a framework is wellknown for mechanical systems with inequality constraints, but we give a more general formulation which applies for instance also to systems with relays in a feedback loop. The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state. A crucial technical tool is the socalled linear complementarity problem (LCP); we introduce various generalizations of this problem. Specific results are obtained for Hamiltonian systems, passive systems, and linear systems.
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. A finite criss ..."
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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.
Basis and Tripartition Identification for Quadratic Programming and Linear Complementarity Problems  From an interior solution to an optimal basis and viceversa
, 1996
"... 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 simplexb ..."
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Cited by 3 (2 self)
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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 simplexbased 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...
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 Matroid programmi ..."
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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.
Principal Pivoting Methods For Linear Complementarity Problems, PCPLCP
"... timization problem min ae c T x + 1 2 x T Qx : Ax b; x 0 oe ; where Q is a positive semidefinite, symmetric matrix, then M = ` 0 A \GammaA T Q ' and q = ` \Gammab c ' : Here M is a positive semidefinite bisymmetric matrix. Bisymmetry means that the matrix has a block diago ..."
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timization problem min ae c T x + 1 2 x T Qx : Ax b; x 0 oe ; where Q is a positive semidefinite, symmetric matrix, then M = ` 0 A \GammaA T Q ' and q = ` \Gammab c ' : Here M is a positive semidefinite bisymmetric matrix. Bisymmetry means that the matrix has a block diagonal structure, and it is the sum of a symmetric block diagonal positive semidefinite, and a skew symmetric block diagonal matrix. Some other classes of solvable LCPs are problems, when M is a ffl P matrix ; ffl sufficient matrix or, equivalently, a P
Chapter 3 SEPARATION PROPERTIES, PRINCIPAL PIVOT TRANSFORMS, CLASSES OF MATRICES
"... In this chapter we present the basic mathematical results on the LCP. Many of these results are used in later chapters to develop algorithms to solve LCPs, and to study the computational complexity of these algorithms. Here, unless stated otherwise, I denotes the unit matrix of order n. M is a given ..."
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In this chapter we present the basic mathematical results on the LCP. Many of these results are used in later chapters to develop algorithms to solve LCPs, and to study the computational complexity of these algorithms. Here, unless stated otherwise, I denotes the unit matrix of order n. M is a given square matrix of order n. In tabular form the LCP (q � M) is w z q I;M q w> 0 � z> 0 � w T z =0 (3:1)
MAPPING AND PRESERVER PROPERTIES OF THE PRINCIPAL PIVOT TRANSFORM
"... Abstract. The principal pivot transform (PPT) is a transformation of a matrix A tantamount to exchanging a fixed set of entries among all of the domainrange vector pairs of A. First in this paper, mapping properties of the PPT applied to certain matrix positivity classes are identified. These class ..."
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Abstract. The principal pivot transform (PPT) is a transformation of a matrix A tantamount to exchanging a fixed set of entries among all of the domainrange vector pairs of A. First in this paper, mapping properties of the PPT applied to certain matrix positivity classes are identified. These classes include the (almost) Pmatrices and (almost) Nmatrices, arising in the linear complementarity problem. Second, a fundamental property of PPTs is proved, namely, that PPTs preserve the rank of the Hermitian part. Third, conditions for the preservation of left eigenspaces by a PPT are determined. Key words. Principal pivot transform; Schur complement; Pmatrix; Nmatrix; Hermitian part; left eigenspace.