Random variational-like inclusion and random proximal operator equation for random fuzzy mappings in Banach spaces

In this paper, we introduce and study a random variational-like inclusion and its corresponding random proximal operator equation for random fuzzy mappings. It is established that the random variational-like inclusion problem for random fuzzy mappings is equivalent to a random fixed point problem. We also establish a relationship between random variational-like inclusion and random proximal operator equation for random fuzzy mappings. This equivalence is used to define an iterative algorithm for solving random proximal operator equation for random fuzzy mappings. Through an example, we show that the random Wardrop equilibrium problem is a special case of the random variational-like inclusion problem for random fuzzy mappings

A System of Generalized Variational Inclusions Involving a New Monotone Mapping in Banach Spaces

We introduce a new monotone mapping in Banach spaces, which is an extension of the -monotone mapping studied by Nazemi (2012), and we generalize the variational inclusion involving the -monotone mapping. Based on the new monotone mapping, we propose a new proximal mapping which combines the proximal mapping studied by Nazemi (2012) with the mapping studied by Lan et al. (2011) and show its Lipschitz continuity. Based on the new proximal mapping, we give an iterative algorithm. Furthermore, we prove the convergence of iterative sequences generated by the algorithm under some appropriate conditions. Our results improve and extend corresponding ones announced by many others

Iterative methods for approximating solutions of certain optimization problems and fixed points problems.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.Abstract available in PDF file

Well-Posedness Properties In Variational Analysis And Its Applications

This dissertation focuses on the study and applications of some significant properties in well-posedness and sensitivity analysis, among which the notions of uniform metric regularity , higher-order metric subregularity and its strong subregularity counterpart play an essential role in modern variational analysis. We derived verifiable sufficient conditions and necessary conditions for those notions in terms of appropriate generalized differential as well as geometric constructions of variational analysis. Concrete examples are provided to illustrate the behavior and compare the results. Optimality conditions of parametric variational systems (PVS) under equilibrium constraints are also investigated via the terms of coderivatives. We derived necessary optimality and suboptimality conditions for various problems of constrained optimization and equilibria such as MPECs with amenable/full rank potentials and EPECs with closed preferences in finite-dimensional spaces

Challenges in Optimal Control of Nonlinear PDE-Systems

The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered

A study of optimization problems and fixed point iterations in Banach spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF

Iterative algorithms for approximating solutions of variational inequality problems and monotone inclusion problems.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban, 2017.In this work, we introduce and study an iterative algorithm independent of the operator norm for approximating a common solution of split equality variational inequality prob- lem and split equality xed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a split equality variational inequality problem and the set of solutions of a split equality xed point problem for demicontractive mappings in real Hilbert spaces. We then considered nite families of split equality variational inequality problems and proposed an iterative algorithm for approximating a common solution of this problem and the multiple-sets split equality xed point problem for countable families of multivalued type-one demicontractive-type mappings in real Hilbert spaces. A strong convergence re- sult of the sequence generated by our proposed algorithm to a solution of this problem was also established. We further extend our study from the frame work of real Hilbert spaces to more general p-uniformly convex Banach spaces which are also uniformly smooth. In this space, we introduce an iterative algorithm and prove a strong convergence theorem for approximating a common solution of split equality monotone inclusion problem and split equality xed point problem for right Bregman strongly nonexpansive mappings. Finally, we presented numerical examples of our theorems and applied our results to study the convex minimization problems and equilibrium problems