2,650 research outputs found
Variational Inclusions with General Over-relaxed Proximal Point and Variational-like Inequalities with Densely Pseudomonotonicity
This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a study of a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal point algorithm to estimate the numerical solution of the implicit inclusion problems. In chapter 3 we introduce generalized densely relaxed ƞ - α pseudomonotone operators and generalized relaxed ƞ - α proper quasimonotone operators as well as relaxed ƞ - α quasimonotone operators. Using these generalized monotonicity notions, we establish the existence results for the generalized variational-like inequality in the general setting of Banach spaces. In chapter 4, we use the auxiliary principle technique to introduce a general algorithm for solutions of the densely relaxed pseudomonotone variational-like inequalities. Chapter 5 is the chapter concluding remarks and scope for future work
Iterative algorithms for solutions of nonlinear equations in Banach spaces.
Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
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