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  • 1
    Electronic Resource
    Electronic Resource
    Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems (1999)
    Springer
    Mathematical programming 86 (1999), S. 105-133 
    Details
    ISSN: 1436-4646
    Keywords: Key words: non-interior point method – complementarity problem – smoothing function – homotopy method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract. We propose a class of non-interior point algorithms for solving the complementarity problems(CP): Find a nonnegative pair (x,y)∈ℝ 2n satisfying y=f(x) and x i y i =0 for every i∈{1,2,...,n}, where f is a continuous mapping from ℝ n to ℝ n . The algorithms are based on the Chen-Harker-Kanzow-Smale smoothing functions for the CP, and have the following features; (a) it traces a trajectory in ℝ 3n which consists of solutions of a family of systems of equations with a parameter, (b) it can be started from an arbitrary (not necessarily positive) point in ℝ 2n in contrast to most of interior-point methods, and (c) its global convergence is ensured for a class of problems including (not strongly) monotone complementarity problems having a feasible interior point. To construct the algorithms, we give a homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition, and propose a class of algorithms involving suitable neighborhoods of the trajectory. We also give a sufficient condition on the neighborhoods for global convergence and two examples satisfying it.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s101070050082
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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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  • 3
    Electronic Resource
    Electronic Resource
    Global convergence in infeasible-interior-point algorithms (1994)
    Springer
    Mathematical programming 65 (1994), S. 43-72 
    Details
    ISSN: 1436-4646
    Keywords: Monotone complementarity problem ; Interior-point algorithm ; Potential reduction algorithm ; Infeasibility ; Global convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581689
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  • 4
    Electronic Resource
    Electronic Resource
    A little theorem of the bigℳ in interior point algorithms (1993)
    Springer
    Mathematical programming 59 (1993), S. 361-375 
    Details
    ISSN: 1436-4646
    Keywords: Interior point algorithm ; big ℳ ; linear program ; convex program ; complementarity problem ; potential reduction algorithm ; self-dual linear program
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an artificial problem having a known interior feasible solution from which we start the algorithm. The artificial problem involves a constantℳ (or constants) which we need to choose large enough to ensure the equivalence between the artificial problem and the original problem. Theoretically, we can always assign a positive number of the order O(2 L ) toℳ in linear cases, whereL denotes the input size of the problem. Practically, however, such a large number is impossible to implement on computers. If we choose too largeℳ, we may have numerical instability and/or computational inefficiency, while the artificial problem withℳ not large enough will never lead to any solution of the original problem. To solve this difficulty, this paper presents “a little theorem of the bigℳ”, which will enable us to find whetherℳ is not large enough, and to updateℳ during the iterations of the algorithm even if we start with a smallerℳ. Applications of the theorem are given to a polynomial-time potential reduction algorithm for positive semi-definite linear complementarity problems, and to an artificial self-dual linear program which has a close relation with the primal—dual interior point algorithm using Lustig's limiting feasible direction vector.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581253
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  • 5
    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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  • 6
    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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  • 7
    Electronic Resource
    Electronic Resource
    A Complexity Analysis of a Smoothing Method Using CHKS-functions for Monotone Linear Complementarity Problems (2000)
    Springer
    Computational optimization and applications 17 (2000), S. 183-201 
    Details
    ISSN: 1573-2894
    Keywords: smoothing method ; complexity bound ; linear complementarity problem ; monotonicity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We consider the standard linear complementarity problem (LCP): Find (x, y) ∈ R 2n such that y = M x + q, (x, y) ≥ 0 and x i y i = 0 (i = 1, 2, ... , n), where M is an n × n matrix and q is an n-dimensional vector. Recently several smoothing methods have been developed for solving monotone and/or P 0 LCPs. The aim of this paper is to derive a complexity bound of smoothing methods using Chen-Harker-Kanzow-Smale functions in the case where the monotone LCP has a feasible interior point. After a smoothing method is provided, some properties of the CHKS-function are described. As a consequence, we show that the algorithm terminates in $$O\left( {\frac{{\gamma ^{ - 6} n}}{{\varepsilon ^6 }}\log \frac{{\gamma ^{ - 2} n}}{{\varepsilon ^2 }}} \right)$$ Newton iterations where $${\bar \gamma }$$ is a number which depends on the problem and the initial point. We also discuss some relationships between the interior point methods and the smoothing methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026550331760
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