The projected Barzilai-Borwein method with fall-back for strictly convex QCQP problems with separable constraints

A variant of the projected Barzilai-Borwein method for solving the strictly convex QCQP problems with separable constraints is presented. The convergence is enforced by a combination of the fall-back strategy and the fixed step-length gradient projection. Using the recent results on the decrease of the convex quadratic function along the projected-gradient path, we prove that the algorithm enjoys the R-linear convergence. The algorithm is plugged into our scalable TFETI based domain decomposition algorithm for the solution of contact problems and its performance is demonstrated on the solution of contact problems, including a frictionless problem and the problems with the isotropic and orthotropic Tresca friction.