Lateef Olakunle Jolaoso

We introduce a double golden-ratio Tseng-type extragradient method (DGR--Tseng) for solving variational inequality problems in real Hilbert spaces. The method exploits two golden-ratio averaging steps while requiring only a single projection per iteration. Weak convergence is established under a solution-oriented condition weaker than monotonicity and pseudomonotonicity, without imposing any finiteness assumption on the solution set. Under a Robinson/Luo--Tseng-type local error-bound condition, we further establish $R$-linear convergence of the distance to the solution set and a $Q$-linear contraction of an associated weighted paired distance, without requiring strong monotonicity, strong pseudomonotonicity, or a singleton solution set. The error bound is weaker than the strong monotonicity-type assumptions commonly used to obtain linear convergence and does not require the solution set to be a singleton. The stepsize is updated adaptively, eliminating the need for prior knowledge of the operator's Lipschitz constant. Numerical experiments on sequence-space and function-space variational inequality problems, together with applications to sparse signal reconstruction and multi-OD urban traffic network equilibrium, demonstrate the computational effectiveness of DGR--Tseng compared with several single golden-ratio methods.

We consider a smooth pessimistic bilevel optimization problem, where the lower-level problem is convex and satisfies the Slater constraint qualification. These assumptions ensure that the Karush-Kuhn-Tucker (KKT) reformulation of our problem is well-defined. We then introduce and study the (i) Scholtes, (ii) Lin and Fukushima, (iii) Kadrani, Dussault and Benchakroun, (iv) Steffensen and Ulbrich, and (v) Kanzow and Schwartz relaxation methods for the KKT reformulation of our pessimistic bilevel program. These relaxations have been extensively studied and compared for mathematical programs with complementarity constraints (MPCCs). To the best of our knowledge, such a study has not been conducted for the pessimistic bilevel optimization problem, which is completely different from an MPCC, as the complementarity conditions are part of the objective function, and not in the feasible set of the problem. After introducing these relaxations, we provide convergence results for global and local optimal solutions, as well as suitable versions of the C- and M-stationarity points of our pessimistic bilevel optimization problem. Numerical results are also provided to illustrate the practical implementation of these relaxation algorithms, as well as some preliminary comparisons.

In this work, we study the split equality variational inclusion and fixed point problems involving Bregman weakly relatively nonexpansive mappings in p -uniformly convex and uniformly smooth real Banach spaces. This research expands upon the current findings that are limited to Hilbert spaces or 2-uniformly convex Banach spaces. We introduce a new simultaneous algorithm with an inertial term to accelerate the rate of convergence of this algorithm and a self-adaptive technique, which is independent of the Lipschitz constant, to approximate a common solution to these problems. This helps to improve the convergence of the method while weakening the conditions on the problem used in existing methods. We also provided some numerical experiments to illustrate the efficiency and accuracy of the method with the non-inertial version.

When the lower-level optimal solution set-valued mapping of a bilevel optimization problem is not single-valued, we are faced with an ill-posed problem, which gives rise to the optimistic and pessimistic bilevel optimization problems, as tractable algorithmic frameworks. However, solving the pessimistic bilevel optimization problem is far more challenging than the optimistic one; hence, the literature has mostly been dedicated to the latter class of the problem. The Scholtes relaxation has appeared to be one of the simplest and most efficient ways to solve the optimistic bilevel optimization problem in its Karush-Kuhn-Tucker (KKT) reformulation or the corresponding more general mathematical program with complementarity constraints (MPCC). Inspired by such a success, this paper studies the potential of the Scholtes relaxation in the context of the pessimistic bilevel optimization problem. To proceed, we consider a pessimistic bilevel optimization problem, where all the functions involved are at least continuously differentiable. Then assuming that the lower-level problem is convex, the KKT reformulation of the problem is considered under the Slater constraint qualification. Based on this KKT reformulation, we introduce the corresponding version of the Scholtes relaxation algorithm. We then construct theoretical results ensuring that the limit of a sequence of global/local optimal solutions (resp. stationary points) of the aforementioned Scholtes relaxation is a global/local optimal solution (resp. stationary point) of the KKT reformulation of the pessimistic bilevel program. The results are accompanied by technical constructions ensuring that the Scholtes relaxation algorithm is well-defined or that the corresponding parametric optimization problem is more tractable. Furthermore, we perform some numerical experiments to assess the performance of the Scholtes relaxation algorithm using various examples. In particular, we study the effectiveness of the algorithm in obtaining solutions that can satisfy the corresponding C-stationarity concept.

In this paper, we suggest a new inertial type parallel iterative algorithm and prove its strong convergence for finding a solution of multiple-sets split equality common fixed point problem for a finite family of demicontractive mappings in real Hilbert spaces. The suggested algorithm does not require prior knowledge of operator norm. We apply our result to study multiple-sets split equality common fixed point problem for a finite family of quasi-pseudocontractive mappings. Further, we also apply our result to solve various split type problems and intensity-modulated radiation therapy. Moreover, we give numerical experiments for supporting our main result and compare it with other existing methods.

The aim of this paper is to study the weak convergence analysis of sequence of iterates generated by a three-operator splitting method of Davis and Yin incorporated with two-step inertial extrapolation for solving monotone inclusion problem involving the sum of two maximal monotone operators and a co-coercive operator in Hilbert spaces. Our results improve on the setbacks observed recently in the literature that one-step inertial Douglas-Rachford splitting method may fail to provide acceleration. Our convergence results also dispense with the summability conditions imposed on inertial parameters and the sequence of iterates assumed in recent results on multi-step inertial methods in the literature. Numerical illustrations from image restoration problem and Smoothly Clipped Absolute Deviation (SCAD) penalty problem are given to show the efficiency and advantage gained by incorporating two-step inertial extrapolation over one-step inertial extrapolation for three-operator splitting method.