  We propose a new modified primal-dual proximal best approximation method for
solving convex not necessarily differentiable optimization problems. The
novelty of the method relies on introducing memory by taking into account
iterates computed in previous steps in the formulas defining current iterate.
To this end we consider projections onto intersections of halfspaces generated
on the basis of the current as well as the previous iterates. To calculate
these projections we are using recently obtained closed-form expressions for
projectors onto polyhedral sets. The resulting algorithm with memory inherits
strong convergence properties of the original best approximation proximal
primal-dual algorithm. Additionally, we compare our algorithm with the original
(non-inertial) one with the help of the so called attraction property defined
below. Extensive numerical experimental results on image reconstruction
problems illustrate the advantages of including memory into the original
algorithm.
