%0 Journal Article
%@ 0022-3239
%A Gnecco, Giorgio
%D 2016
%F eprints:3123
%I Springer
%J Journal of Optimization Theory and Applications
%K Ritz method; Curse of dimensionality; Infinite-dimensional optimization; Approximation schemes; Extended Ritz method
%N 2
%P 488-509
%T On the Curse of Dimensionality in the Ritz Method
%U http://eprints.imtlucca.it/3123/
%V 168
%X It is shown that the classical Ritz method of the calculus of variations suffers from the “curse of dimensionality,” i.e., an exponential growth, as a function of the number of variables, of the dimension a linear subspace needs in order to achieve a desired relative improvement in the accuracy of approximation of the optimal solution value. The proof is constructive and is obtained by exhibiting a family of infinite-dimensional optimization problems for which this happens, namely those with quadratic functional and spherical constraint. The results provide a theoretical motivation for the search of alternative solution methods, such as the so-called “extended Ritz method,” to deal with the curse of dimensionality.
%Z First online: 03 September 2015