Rigorous convergence analysis of alternating variable minimization with multiplier methods for quadratic programming problems with equality constraints

We discuss unique solvability of the equality-constraint quadratic programming problem, establish a class of preconditioned alternating variable minimization with multiplier (PAVMM) methods for iteratively computing its solution, and demonstrate asymptotic convergence property of these PAVMM methods. We also discuss an algebraic derivation of the PAVMM method by making use of matrix splitting, which reveals that the PAVMM method is actually a modified block Gauss–Seidel iteration method for solving the augmented Lagrangian linear system resulting from the weighted Lagrangian function with respect to the equality-constraint quadratic programming problem.