This book is devoted to a detailed study of the subgradient projection method and its variants for convex optimization problems over the solution sets of common fixed point problems and convex feasibility problems. These optimization problems are investigated to determine good solutions obtained by different versions of the subgradient projection algorithm in the presence of sufficiently small computational errors. The use of selected algorithms is highlighted including the Cimmino type subgradient, the iterative subgradient, and the dynamic string-averaging subgradient. All results presented are new. Optimization problems where the underlying constraints are the solution sets of other problems, frequently occur in applied mathematics. The reader should not miss the section in Chapter 1 which considers some examples arising in the real world applications. The problems discussed have an important impact in optimization theory as well. The book will be useful for researches interested in the optimization theory and its applications. Preface Contents 1 Introduction 1.1 Subgradient Projection Method 1.2 Fixed Point Subgradient Algorithms 1.3 Proximal Point Subgradient Algorithm 1.4 Cimmino Subgradient Projection Algorithm 1.5 Examples 2 Fixed Point Subgradient Algorithm 2.1 Common Fixed Point Problems 2.2 The Cimmino Subgradient Algorithm 2.3 Two Auxiliary Results 2.4 The First Result for the Cimmino Subgradient Algorithm 2.5 The Second Result for the Cimmino Subgradient Algorithm 2.6 The Iterative Subgradient Algorithm 2.7 Auxiliary Results 2.8 Convergence Results for the Iterative Subgradient Algorithm 2.9 Dynamic String-Averaging Subgradient Algorithm 2.10 Auxiliary Results 2.11 The First Theorem for the Dynamic String-Averaging Subgradient Algorithm 2.12 The Second Theorem for the Dynamic String-Averaging Subgradient Algorithm 3 Proximal Point Subgradient Algorithm 3.1 Preliminaries 3.2 Auxiliary Results 3.3 The First Result for the Cimmino Proximal Point Subgradient Algorithm 3.4 The Second Result for the Cimmino Proximal Point Subgradient Algorithm 3.5 The Iterative Proximal Point Subgradient Algorithm 3.6 The First Theorem for the Iterative Proximal Point Subgradient Algorithm 3.7 The Second Theorem for the Iterative Proximal Point Subgradient Algorithm 3.8 Dynamic String-Averaging Proximal Point Subgradient Algorithm 3.9 Auxiliary Results 3.10 The First Theorem for the DSA Proximal Point Subgradient Algorithm 3.11 The Second Theorem for the DSA Subgradient Algorithm 4 Cimmino Subgradient Projection Algorithm 4.1 Preliminaries 4.2 Cimmino Subgradient Projection Algorithm 4.3 Two Convergence Results 4.4 The Third and Fourth Convergence Results 5 Iterative Subgradient Projection Algorithm 5.1 Preliminaries 5.2 Auxiliary Results 5.3 The Main Results 6 Dynamic String-Averaging Subgradient Projection Algorithm 6.1 Preliminaries 6.2 The Basic Auxiliary Result 6.3 The Main Results 7 Fixed Point Gradient Projection Algorithm 7.1 Preliminaries 7.2 The Basic Lemma 7.3 An optimization problem 8 Cimmino Gradient Projection Algorithm 8.1 Preliminaries 8.2 Cimmino Type Gradient Algorithm 8.3 The Basic Lemma 8.4 Main Results 9 A Class of Nonsmooth Convex Optimization Problems 9.1 Preliminaries 9.2 Approximate Solutions of Problems on Bounded Sets 9.3 Approximate Solutions of Problems on Unbounded Sets 9.4 Convergence to the Set of Minimizers 9.5 Auxiliary Results on the Convergence of Infinite Products 9.6 Auxiliary Results 9.7 Proof of Theorem 9.1 9.8 Proof of Theorem 9.2 9.9 Proof of Theorem 9.3 9.10 Proof of Theorem 9.4 9.11 Proof of Theorem 9.5 9.12 Proof of Theorem 9.6 9.13 Proof of Theorem 9.7 9.14 Proof of Theorem 9.8 9.15 Proof of Theorem 9.9 10 Zero-Sum Games with Two Players 10.1 Preliminaries and an Auxiliary Result 10.2 Zero-Sum Games on Bounded Sets 10.3 The First Main Result 10.4 The Second Main Result References Index
