Modified mildly inertial subgradient extragradient method for solving pseudomonotone equilibrium problems and nonexpansive fixed point problems

This paper presents and examines a newly improved linear technique for solving the equilibrium problem of a pseudomonotone operator and the fixed point problem of a nonexpansive mapping within a real Hilbert space framework. The technique relies two modified mildly inertial methods and the subgradient extragradient approach. In addition, it can be viewed as an advancement over the previously known inertial subgradient extragradient approach. Based on common assumptions, the algorithm's weak convergence has been established. Finally, in order to confirm the efficiency and benefit of the proposed algorithm, we present a few numerical experiments

A new method with regularization for solving split variational inequality problem in real Hilbert spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.The concept of the optimization problem, fixed point theory and its application constitute the nucleus of nonlinear analysis, which is a major branch of mathematics. Optimization theory, fixed point theory and its applications have a wide range of application in practically every field of science, particularly mathematical sciences. The theory of optimization and fixed point have received great attention from authors around the world and these areas will continue to receive such great attention. The theory has been well developed by well-known researchers in these areas. However, there are still a lot of work to be done. The goal of this thesis is to advance the theory of optimization and fixed point in the framework of Hilbert and Banach spaces. The substance of this thesis is separated into two parts. The research efforts of the first part of this thesis (Chapter 3 to Chapter 6) has to do with introducing some new iterative methods for approximating the solution of a variational inequality problems, split variational inequality problems, equilibrium problems, split monotone variational inclusion problem, split generalized mixed equilibrium problem and fixed point problems in the framework of a Hilbert and Banach spaces. In addition, we introduce a new class of bilevel problem in the framework of real Hilbert spaces and a new regularization technique, and inertial terms for approximating solutions of split bilevel variational inequality problems. Furthermore, we establish that the proposed iterative methods converges strongly to the solution of the aforementioned problems as the case may be. Then, we present some numerical experiments to show the efficiency and applicability of our proposed methods in comparison with other state-of-the-art iterative methods in the literature. The second part (Chapter 7) of this thesis deals with developing iterative algorithms and introducing some nonlinear mappings in the framework of the Hilbert and Banach spaces. First, we present a modified (improved) generalized Miteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we present some fixed point results for a general class of nonexpansive mappings in the framework of the Banach space and also proposed a new iterative scheme for approximating the fixed point of this class of mappings in the framework of uniformly convex Banach spaces. Finally, we apply our convergence results to certain optimization problems, integral equations, and we present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature

Enhanced intelligent water drops algorithm for multi-depot vehicle routing problem.

The intelligent water drop algorithm is a swarm-based metaheuristic algorithm, inspired by the characteristics of water drops in the river and the environmental changes resulting from the action of the flowing river. Since its appearance as an alternative stochastic optimization method, the algorithm has found applications in solving a wide range of combinatorial and functional optimization problems. This paper presents an improved intelligent water drop algorithm for solving multi-depot vehicle routing problems. A simulated annealing algorithm was introduced into the proposed algorithm as a local search metaheuristic to prevent the intelligent water drop algorithm from getting trapped into local minima and also improve its solution quality. In addition, some of the potential problematic issues associated with using simulated annealing that include high computational runtime and exponential calculation of the probability of acceptance criteria, are investigated. The exponential calculation of the probability of acceptance criteria for the simulated annealing based techniques is computationally expensive. Therefore, in order to maximize the performance of the intelligent water drop algorithm using simulated annealing, a better way of calculating the probability of acceptance criteria is considered. The performance of the proposed hybrid algorithm is evaluated by using 33 standard test problems, with the results obtained compared with the solutions offered by four well-known techniques from the subject literature. Experimental results and statistical tests show that the new method possesses outstanding performance in terms of solution quality and runtime consumed. In addition, the proposed algorithm is suitable for solving large-scale problems

The parameter values for IWD procedures.

The parameter values for IWD procedures.</p

Mean rank of the IWD-ASA algorithm with other methods.

Mean rank of the IWD-ASA algorithm with other methods.</p

Gaps between IWD-SA and literature techniques.

<p>Gaps between IWD-SA and literature techniques.</p

Convergence trends of IWD, IWD-SA, and IWD-ASA for the P04 instance.

Convergence trends of IWD, IWD-SA, and IWD-ASA for the P04 instance.</p

The parameter values for SA procedures.

The parameter values for SA procedures.</p

Average ranking returned by Friedman’s non-parametric test for the 33 MDVRP instances.

Average ranking returned by Friedman’s non-parametric test for the 33 MDVRP instances.</p

Mean rank of the IWD-SA algorithm with other methods.

<p>Mean rank of the IWD-SA algorithm with other methods.</p