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  • 1
    Electronic Resource
    Electronic Resource
    Determining basic variables of optimal solutions in Karmarkar's new LP algorithm (1986)
    Springer
    Algorithmica 1 (1986), S. 499-515 
    Details
    ISSN: 1432-0541
    Keywords: Linear program ; Karmarkar's algorithm ; Optimal basis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper establishes a sufficient condition for a variable of a linear program to be positive at all optimal solutions. A numerical test using the condition is incorporated into Karmarkar's new LP algorithm to determine columns of optimal basis. Experimental results on the test are also reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01840459
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    Paper (German National Licenses)
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  • 2
    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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  • 3
    Electronic Resource
    Electronic Resource
    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
    Keywords: Potential reduction algorithm ; linear complementarity problem ; interior point algorithm ; Karmarkar's algorithm ; path of centers ; central trajectory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01594942
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    Paper (German National Licenses)
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