Weak and strong convergences of the generalized penalty Forward–Forward and Forward–Backward splitting algorithms for solving bilevel hierarchical pseudomonotone equilibrium problems

ABSTRACT

In this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward–Forward algorithm (PFFSA) and the Forward–Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field.

Keywords:

• Pseudomonotone bifunction
• hierarchical equilibrium problem
• penalty Forward–Forward splitting algorithm
• penalty Forward–Backward splitting algorithm
• weak convergence
• weak ergodic convergence
• strong convergence

AMS CLASSIFICTION:

• 47H05
• 49M30
• 65K05
• 90C25
• 90C47

Acknowledgments

The authors wish to thank two anonymous Referees and the Editor for the careful reading, useful remarks and very nice constructive comments which motivated them to improve the quality and the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the National Centre for Scientific and Technical Research (CNRST Morocco) [grant number URAC01].