Available online at www.sciencedirect.com ScienceDirect Mathematics and Computers in Simulation 145 (2018) 79–89 www.elsevier.com/locate/matcom Original articles The projected Barzilai–Borwein method with fall-back for strictly convex QCQP problems with separable constraints Lukáš Pospíšil ∗ , Zdenˇ ek Dostál VŠB-Technical University of Ostrava, Tˇ r 17. listopadu 15, CZ-70833 Ostrava, Czech Republic Received 4 November 2014; received in revised form 6 July 2015; accepted 6 October 2017 Available online 23 October 2017 Abstract 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. c ⃝ 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved. Keywords: Quadratic programming; Separable constraints; QCQP; Barzilai–Borwein with fall-back 1. Introduction We consider a problem of minimizing the strictly convex quadratic function subject to separable convex constraints, i.e., we are looking for ˆ x = arg min x ∈Ω f (x ), f (x ) = 1 2 x T Ax − b T x , (1) where A ∈ R n×n is a symmetric positive definite (SPD) matrix, b, x ∈ R n , and Ω ⊆ R n is a non-empty closed convex set defined by convex differentiable functions h i : R ℓ i → R, i = 1,..., m, so that Ω = Ω 1 ×···× Ω m ∗ Corresponding author. E-mail addresses: lukas.pospisil@vsb.cz (L. Posp´ ıˇ sil), zdenek.dostal@vsb.cz (Z. Dost´ al). https://doi.org/10.1016/j.matcom.2017.10.003 0378-4754/ c ⃝ 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.