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  • Digitale Medien  (4)
  • Linear complementarity problem  (2)
  • Potential reduction algorithm  (2)
  • 1
    Digitale Medien
    Digitale Medien
    A polynomial-time algorithm for a class of linear complementarity problems (1989)
    Springer
    Mathematical programming 44 (1989), S. 1-26 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Linear complementarity problem ; polynomial-time algorithm ; path of centers ; Karmarkar's algorithm
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01587074
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  • 2
    Digitale Medien
    Digitale Medien
    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
    Schlagwort(e): Linear complementarity problem ; ellipsoid ; interior point algorithm ; path following algorithm
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01582266
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  • 3
    Digitale Medien
    Digitale Medien
    An $$O(\sqrt n L)$$ iteration potential reduction algorithm for linear complementarity problems (1991)
    Springer
    Mathematical programming 50 (1991), S. 331-342 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Potential reduction algorithm ; linear complementarity problem ; interior point algorithm ; Karmarkar's algorithm ; path of centers ; central trajectory
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract This paper proposes an interior point algorithm for a positive semi-definite linear complementarity problem: find an (x, y)∈ℝ 2n such thaty=Mx+q, (x,y)⩾0 andx T y=0. The algorithm reduces the potential function $$f(x,y) = (n + \sqrt n )\log x^T y - \sum\limits_{i = 1}^n {\log x_i y_i } $$ by at least 0.2 in each iteration requiring O(n 3) arithmetic operations. If it starts from an interior feasible solution with the potential function value bounded by $$O(\sqrt n L)$$ , it generates, in at most $$O(\sqrt n L)$$ iterations, an approximate solution with the potential function value $$ - O(\sqrt n L)$$ , from which we can compute an exact solution in O(n 3) arithmetic operations. The algorithm is closely related with the central path following algorithm recently given by the authors. We also suggest a unified model for both potential reduction and path following algorithms for positive semi-definite linear complementarity problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01594942
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  • 4
    Digitale Medien
    Digitale Medien
    Global convergence in infeasible-interior-point algorithms (1994)
    Springer
    Mathematical programming 65 (1994), S. 43-72 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Monotone complementarity problem ; Interior-point algorithm ; Potential reduction algorithm ; Infeasibility ; Global convergence
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract This paper presents a wide class of globally convergent interior-point algorithms for the nonlinear complementarity problem with a continuously differentiable monotone mapping in terms of a unified global convergence theory given by Polak in 1971 for general nonlinear programs. The class of algorithms is characterized as: Move in a Newton direction for approximating a point on the path of centers of the complementarity problem at each iteration. Starting from a strictly positive but infeasible initial point, each algorithm in the class either generates an approximate solution with a given accuracy or provides us with information that the complementarity problem has no solution in a given bounded set. We present three typical examples of our interior-point algorithms, a horn neighborhood model, a constrained potential reduction model with the use of the standard potential function, and a pure potential reduction model with the use of a new potential function.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01581689
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