Browsing by Subject "Newton's method"

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    A new finite continuation algorithm for linear programming
    (Society for Industrial and Applied Mathematics, 1996) Madsen, K.; Nielsen, H. B.; Pınar, M. Ç.
    We describe a new finite continuation algorithm for linear programming. The dual of the linear programming problem with unit lower and upper bounds is formulated as an f\ minimization problem augmented with the addition of a linear term. This nondifferentiable problem is approximated by a smooth problem. It is shown that the minimizers of the smooth problem define a family of piecewise-linear paths as a function of a smoothing parameter. Based on this property, a finite algorithm that traces these paths to arrive at an optimal solution of the linear program is developed. The smooth problems are solved by a Newton-type algorithm. Preliminary numerical results indicate that the new algorithm is promising.
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    Newton's method for linear inequality systems
    (Elsevier, 1998) Pınar, M. Ç.
    We describe a modified Newton type algorithm for the solution of linear inequality systems in the sense of minimizing the ℓ2 norm of infeasibilities. Finite termination is proved, and numerical results are given. © 1998 Elsevier Science B.V.
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    On Newton's method for Huber's robust M-estimation problems in linear regression
    (Springer Netherlands, 1998) Chen, B.; Pınar, M. Ç.
    The Newton method of Madsen and Nielsen (1990) for computing Huber's robust M-estimate in linear regression is considered. The original method was proved to converge finitely for full rank problems under some additional restrictions on the choice of the search direction and the step length in some degenerate cases. It was later observed that these requirements can be relaxed in a practical implementation while preserving the effectiveness and even improving the efficiency of the method. In the present paper these enhancements to the original algorithm are studied and the finite termination property of the algorithm is proved without any assumptions on the M-estimation problems.