On the construction of smoothing splines by quadratic programming (Q2718813)

The problem of minimization of the smoothing functional NEWLINE\[NEWLINEJ(f)=\int^b_a(f^{(q)}_{(t)})^2dtNEWLINE\]NEWLINE in the Sobolev space \(W^q_2[a,b]\) under the inequality constraints NEWLINE\[NEWLINE|f(t_i)-z_i|\leq \varepsilon_i,\quad (i=1,2,\dots,n)NEWLINE\]NEWLINE is analyzed. It is proved that this is equivalent to a problem of quadratic programming with a positive semidefinite matrix. A modification of the simplex method appropriate for the conditions of the problem which was suggested by \textit{Ph. Wolfe} [Econometrica 27, 382-398 (1959; Zbl 0103.37603)] and \textit{V. A. Daugaret} [Zh. Vychisl. Mat. Mat. Fiz. 21, 504-508 (1981; Zbl 0507.90070)] is applied. The more important result is the theorem 3.1.