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A new stable quadratic minimization method on a half-space is presented. In case of normally solvable operators this method outperforms approximate solutions having the same optimal order accuracy as earlier methods for unconstrained problems. | Yugoslav Journal of Operations Research 14 (2004), Number 1, 19-26 ON THE ACCURACY OF REGULARIZED SOLUTIONS TO QUADRATIC MINIMIZATION PROBLEMS ON A HALFSPACE, IN CASE OF A NORMALLY SOLVABLE OPERATOR* I. KRNIĆ, O. OBRADOVIĆ, Department of Mathematics, University of Montenegro Podgorica, Serbia and Montenegro M.M. POTAPOV Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia Received: September 2003 / Accepted: February 2004 Abstract: A new stable quadratic minimization method on a half-space is presented. In case of normally solvable operators this method outperforms approximate solutions having the same optimal order accuracy as earlier methods for unconstrained problems. Keywords: Quadratic functional, linear constraints, optimization, regularization. 1. STATE OF THE PROBLEM. STRUCTURE OF THE OPTIMAL SOLUTION. In this paper we consider a quadratic minimization problem with a linear constraint: J (u ) = 1 || Au − f ||2F → inf, 2 c, u H ≤r (1) Here H and F are Hilbert spaces, f ∈ F , c ∈ H , (c ≠ 0) and r ∈ R . Operator A ∈ L ( H → F ) is a linear bounded operator which is normally solvable, i.e. * 1991 Mathematics Subject Classification, 65J10, 65J20. 20 I. Krnić, O. Obradović, M. Potapov / On the Accuracy of Regularized Solutions R ( A) = R ( A) , (2) where R ( A) = {v ∈ F | ∃u ∈ H , v = Au} is the range of the operator A. We also assume that, instead of exact data A, f, and c we are given their approximations Aµ ∈ L ( H → F ), fδ ∈ F , cσ ∈ H , and corresponding error levels µ , δ ,σ > 0 : || A − Aµ || ≤ µ , || f − fδ || ≤ δ , || c − cσ || ≤ σ . (3) The same problem (1) with perturbed data (3), but without a normal solvability assumption (2) was considered earlier in [1] and it was shown that the accuracy order of the output approximations generated by the algorithm from [1] was bounded by the value 1 ( µ + δ + σ ) 2 . On the other hand, for unconstrained minimization problems 1 || Au − f ||2 → inf, u ∈ H
