Focusing on order analysis and deep learning, this book explores crucial links to optimization, including nonlinear, vector, and set optimization. It provides a comprehensive review of foundational concepts, followed by two main sections that delve deeper into these topics, making it a valuable resource for understanding the interplay between deep learning and optimization techniques.
In vector optimization, the focus is on identifying optimal elements—such as minimal or weakly minimal elements—within a nonempty subset of a partially ordered linear space. The challenge of finding at least one of these optimal elements, if they exist, is termed a vector optimization problem. These problems extend beyond mathematics, appearing in fields like engineering and economics. They manifest in functional analysis (e.g., Hahn-Banach theorem, Bishop-Phelps lemma), multiobjective programming, multi-criteria decision making, statistics (Bayes solutions, minimal covariance matrices), approximation theory, and cooperative game theory, including optimal control problems. Recently, vector optimization has evolved to include set-valued maps, leading to a new research area known as set optimization, which has significant implications for variational inequalities and optimization with multivalued data. The origins of vector optimization trace back to F. Y. Edgeworth and V. Pareto, who laid the groundwork for multiobjective optimization concepts. The field gained momentum with the influential paper by H. W. Kuhn and A. W. Tucker in 1951, and since the late 1960s, research in vector optimization has intensified.
This book serves as an introductory text to optimization theory in normed spaces and covers all areas of nonlinear optimization. It presents fundamentals with particular emphasis on the application to problems in the calculus of variations, approximation and optimal control theory. The reader is expected to have a basic knowledge of linear functional analysis.