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  • 1
    Electronic Resource
    Electronic Resource
    On the existence and uniqueness of solutions in nonlinear complementarity theory (1977)
    Springer
    Mathematical programming 12 (1977), S. 110-130 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A complementarity problem is said to be globally uniquely solvable (GUS) if it has a unique solution, and this property will not change, even if any constant term is added to the mapping generating the problem. A characterization of the GUS property which generalizes a basic theorem in linear complementarity theory is given. Known sufficient conditions given by Cottle, Karamardian, and Moré for the nonlinear case are also shown to be generalized. In particular, several open questions concerning Cottle's condition are settled and a new proof is given for the sufficiency of this condition. A simple characterization for the two-dimensional case and a necessary condition for then-dimensional case are also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593774
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A PL homotopy for finding all the roots of a polynomial (1979)
    Springer
    Mathematical programming 16 (1979), S. 37-62 
    Details
    ISSN: 1436-4646
    Keywords: Roots of Polynomials ; Fixed Point Computing Methods ; Complementary Pivoting Methods ; Piecewise Linear Homotopy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents a constructive method which gives, for any polynomialF(Z) of the degreen, approximate values of all the roots ofF(Z).. The point of the method is on the use of a piecewise linear function $$\bar H$$ (Z, t) which approximates a homotopyH(Z, t) betweenF(Z) and a polynomialG(Z) of the degreen withn known simple roots. It is shown that the set of solutions to $$\bar H$$ (Z, t) = 0 includesn distinct paths,m of which converges to a root ofF(Z) if and only if the root has the multiplicitym. Starting from givenn roots ofG(Z), a complementary pivot algorithm generates thosen paths.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582093
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods (1982)
    Springer
    Mathematical programming 24 (1982), S. 177-215 
    Details
    ISSN: 1436-4646
    Keywords: Variable Dimension Algorithm ; Fixed Point ; Subdivided Manifold ; Nonlinear Equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman. We introduce a new concept ‘primal—dual pair of subdivided manifolds’ and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms. Our basic model furnishes interpretations to several existing methods: Lemke's algorithm for the linear complementarity problem, its extension to the nonlinear complementarity problem, a variable dimension algorithm on conical subdivisions and Merrill's algorithm. We shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us. A method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585103
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    A unification of the existence theorems of the nonlinear complementarity problem (1975)
    Springer
    Mathematical programming 9 (1975), S. 257-277 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The nonlinear complementarity problem is the problem of finding a point x in the n-dimensional Euclidean space,R n , such that x ⩾ 0, f(x) ⩾ 0 and 〈x,f(x)∼ = 0, where f is a nonlinear continuous function fromR n into itself. Many existence theorems for the problem have been established in various ways. The aim of the present paper is to treat them in a unified manner. Eaves's basic theorem of complementarity is generalized, and the generalized theorem is used as a unified framework for several typical existence theorems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01681350
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    A modification of todd's triangulationJ 3 (1978)
    Springer
    Mathematical programming 15 (1978), S. 223-227 
    Details
    ISSN: 1436-4646
    Keywords: Triangulations ; Fixed Point Computing Methods ; Complementary Pivoting Methods ; Piecewise Linear Homotopy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract When we apply the fixed point computing method to mappings which are affine in some variables, we show that, to generate a sequence which converges to a fixed point, the mesh size need not be decreased in these coordinates. This paper modifies the triangulationJ 3 with continuous refinement of mesh size to a triangulation $$\bar J_3$$ such that the mesh size of $$\bar J_3$$ in some given coordinates is constant and the mesh size in the other coordinates shrinks to zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01609022
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    Paper (German National Licenses)
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