Projection algorithms and monotone operators
Jonathan Michael Borwein, Heinz H. Bauschke · 1996
This thesis consists of two parts. In Part I, projection algorithms for solving convex feasibility problems in Hilbert space are studied. Powerful techniques from Convex Analysis are employed within a very general framework that covers and extends many well-known results. Ostensibly different looking conditions sufficient for linear convergence are shown to be special instances of regularity--a concept new in this context. Numerous examples, including subgradient algorithms, are presented. Several notions of monotonicity of operators on Banach spaces are analyzed in Part II. Utilizing Convex and Functional Analysis, it is shown that for a bounded linear positive semi-definite operator, all these monotonicities coincide with the monotonicity of the conjugate operator. Moreover, monotonicity of the conjugate operator is automatic in many classical Banach spaces but not in spaces containing a complemented copy of the space of absolutely convergent sequences.