Browsing by Subject "Huber's M-estimator"
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Item Open Access Bound constrained quadratic programming via piecewise quadratic functions(Springer-Verlag, 1999) Madsen, K.; Nielsen, H. B.; Pınar, M. Ç.We consider the strictly convex quadratic programming problem with bounded variables. A dual problem is derived using Lagrange duality. The dual problem is the minimization of an unconstrained, piecewise quadratic function. It involves a lower bound of λ1 , the smallest eigenvalue of a symmetric, positive definite matrix, and is solved by Newton iteration with line search. The paper describes the algorithm and its implementation including estimation of λ1, how to get a good starting point for the iteration, and up- and downdating of Cholesky factorization. Results of extensive testing and comparison with other methods for constrained QP are given. © Springer-Verlag 1999.Item Open Access Duality in robust linear regression using Huber's M-estimator(Elsevier, 1997-07) Pınar, M. Ç.The robust linear regression problem using Huber's piecewise-quadratic M-estimator function is considered. Without exception, computational algorithms for this problem have been primal in nature. In this note, a dual formulation of this problem is derived using Lagrangean duality. It is shown that the dual problem is a strictly convex separable quadratic minimization problem with linear equality and box constraints. Furthermore, the primal solution (Huber's M-estimate) is obtained as the optimal values of the Lagrange multipliers associated with the dual problem. As a result, Huber's M-estimate can be computed using off-the-shelf optimization software.Item Open Access A finite continuation algorithm for bound constrained quadratic programming(Society for Industrial and Applied Mathematics Publications, 1998) Madsen, K.; Nielsen, H. B.; Pınar, M. C.The dual of the strictly convex quadratic programming problem with unit bounds is posed as a linear ℓ1 minimization problem with quadratic terms. A smooth approximation to the linear ℓ1 function is used to obtain a parametric family of piecewise-quadratic approximation problems. The unique path generated by the minimizers of these problems yields the solution to the original problem for finite values of the approximation parameter. Thus, a finite continuation algorithm is designed. Results of extensive computational experiments are reported.Item Open Access Linear huber M-estimator under ellipsoidal data uncertainty(Springer, 2002) Pınar, M. Ç.The purpose of this note is to present a robust counterpart of the Huber estimation problem in the sense of Ben-Tal and Nemirovski when the data elements are subject to ellipsoidal uncertainty. The robust counterparts are polynomially solvable second-order cone programs with the strong duality property. We illustrate the effectiveness of the robust counterpart approach on a numerical example.