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This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. Mainly it is concerned with the existence of solutions of a given problem in these classes, whenever for example standard compacity hypothesis is not present. Thus it addresses the central problem of handling unbounded situations. This book will be useful to advanced graduate students, researchers, and practitioners in the fields of optimization theory, nonlinear programming, and applied mathematical sciences.