Penalty schemes with inertial effects for monotone inclusion problems

Abstract

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function.

Keywords:

• Maximally monotone operator
• Fitzpatrick function
• resolvent
• Lipschitz continuous operator
• forward–backward-forward algorithm
• subdifferential
• Fenchel conjugate

Notes

No potential conflict of interest was reported by the authors.

Dedicated to Professor Franco Giannessi on the occasion of his 80th birthday and to Professor Diethard Pallaschke on the occasion of his 75th birthday.

Additional information

Funding

This work was supported by FWF (Austrian Science Fund), Lise Meitner Programme, project [M 1682-N25].