  We consider a linear iterative solver for large scale linearly constrained
quadratic minimization problems that arise, for example, in optimization with
PDEs. By a primal-dual projection (PDP) iteration, which can be interpreted and
analysed as a gradient method on a quotient space, the given problem can be
solved by computing sulutions for a sequence of constrained surrogate problems,
projections onto the feasible subspaces, and Lagrange multiplier updates. As a
major application we consider a class of optimization problems with PDEs, where
PDP can be applied together with a projected cg method using a block triangular
constraint preconditioner. Numerical experiments show reliable and competitive
performance for an optimal control problem in elasticity.
