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  • Articles: DFG German National Licenses  (4)
  • Electronic Resource  (4)
  • Complementary Pivoting Methods  (2)
  • Linear complementarity problem  (2)
  • 1
    Electronic Resource
    Electronic Resource
    A polynomial-time algorithm for a class of linear complementarity problems (1989)
    Springer
    Mathematical programming 44 (1989), S. 1-26 
    Details
    ISSN: 1436-4646
    Keywords: Linear complementarity problem ; polynomial-time algorithm ; path of centers ; Karmarkar's algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x ≥ 0,y ≥ 0,x i y i = 0 (i = 1, 2,⋯,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n 3 L) arithmetic operations by tracing the path of centers,{(x, y) ∈ S: x i y i =μ (i = 1, 2,⋯,n) for some μ 〉 0} of the feasible regionS = {(x, y) ≥ 0:y = Mx + q}, whereL denotes the size of the input data of the problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01587074
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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
    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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  • 4
    Electronic Resource
    Electronic Resource
    Ellipsoids that contain all the solutions of a positive semi-definite linear complementarity problem (1990)
    Springer
    Mathematical programming 48 (1990), S. 415-435 
    Details
    ISSN: 1436-4646
    Keywords: Linear complementarity problem ; ellipsoid ; interior point algorithm ; path following algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper deals with the LCP (linear complementarity problem) with a positive semi-definite matrix. Assuming that a strictly positive feasible solution of the LCP is available, we propose ellipsoids each of which contains all the solutions of the LCP. We use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the LCP. We can apply the lower bound to test whether a given variable is positive over the solution set of the LCP. That is, if the lower bound is positive, we know that the variable is positive over the solution set of the LCP; hence, by the complementarity condition, its complement is zero. In this case we can eliminate the variable and its complement from the LCP. We also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path-following algorithm proposed by the authors for the LCP. If the LCP has a unique non-degenerate solution, the lower bound and the upper bound for the solution, computed at each iteration of the path-following algorithm, both converge to the solution of the LCP.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582266
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    Paper (German National Licenses)
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