A forward–backward penalty scheme with inertial effects for monotoneinclusions. Applications to convex bilevel programming

摘要

We investigate a forward–backward splitting algorithm of penalty type with inertialeffects for finding the zeros of the sum of a maximally monotone operator and a cocoerciveone and the convex normal cone to the set of zeroes of an another cocoercive operator.Weak ergodic convergence is obtained for the iterates, provided that a condition expressedvia the Fitzpatrick function of the operator describing the underlying set of the normalcone is verified. Under strong monotonicity assumptions, strong convergence for thesequence of generated iterates is proved. As a particular instance we consider a convexbilevel minimization problem including the sum of a non-smooth and a smooth function inthe upper level and another smooth function in the lower level. We show that in thiscontext weak non-ergodic and strong convergence can be also achieved under inf-compactnessassumptions for the involved functions.