Abstract
ABSTRACTThis paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar's proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar's PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm is a (large-step) relaxed hybrid proximal extragradient (r-HPE) method of multipliers, which combines Rockafellar's PMM with the r-HPE method. We obtain pointwise and ergodic global convergence rates at the price of solving, at each iteration, quadratic quadratically constrained convex programming problems. These convergence rates are superior to the corresponding pointwise and ergodic currently known for standard proximal-point methods, thanks to the incorporation of second-order information. To the best of our knowledge, this is the first time that the above mentioned rates and results are obtained for solving the smooth convex programming problems with inequality constraints.
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