Iterative Methods for Quadratic Programming

In the present paper, we consider medium size quadratic programs $$ \frac{1}{2}{x^{T}}Cx + {p^{T}}x = \min !\quad {\text{s}}{\text{.t}}{\text{.}}\quad x \in G \hfill \\ G: = \{ x \in {R^{{\text{n}}}}\left| {{{({A^{T}}x - b)}^{i}}} \right. \leqslant 0,\;i \in I\} \ne \emptyset ,\quad I: = \{ 1, \ldots ,m\} \hfill \\ $$ (1) , where C symmetric and positive definite. In contrast to finite methods for solving (1) (e.g. [3,4, 7, 8, 9]), we consider infinite iteration processes, for instance penalty and Newton-type methods, and propose a general termination technique for these processes. In this way, it can be ensured under mild assumptions that after a finite number of steps the optimal solution is obtained, even if the normals of the active constraints are not linearly independent [2].