  For optimization problems with linear equality constraints, we prove that the
(1,1) block of the inverse KKT matrix remains unchanged when projected onto the
nullspace of the constraint matrix. We develop reduced compact representations
of the limited-memory inverse BFGS Hessian to compute search directions
efficiently when the constraint Jacobian is sparse. Orthogonal projections are
implemented by a sparse QR factorization or a preconditioned LSQR iteration. In
numerical experiments two proposed trust-region algorithms improve in
computation times, often significantly, compared to previous implementations of
related algorithms and compared to IPOPT.
