Modified Lagrangians and Monotone Maps in OptimizationThis translation of the important Russian text covers the theory and computational methods of modified Lagrangian functions (MLFs)—a new branch of mathematical programming used to solve optimization problems. Providing a thorough analysis for both traditional convex programming and monotone maps, the book shows the advantages of MLFs over classical Lagrangian functions in such practical applications as numerical algorithms, economic modeling, decomposition, and nonconvex local constrained optimization. Following an overview of convex analysis, the authors introduce MLFs through the more general formalism of weak modified Lagrangian functions (WMLFs). They use the two concepts to develop a theory of duality supported by examples of elementary economic models. Also examined are the benefits of MLFs in the application of dual methods in linear programming and in problems with inconsistent constraints. This is the first volume in which monotone maps are treated broadly, in line with their growing importance in optimization and mathematical economics. Two chapters on monotone maps cover pointtoset maps, propose modifications that would achieve a pointtopoint map with improved properties, show how to arrive at new MLF constructions, and detail decomposition methods for convex programming. A chapter on the saddle gradient method covers convergence properties exhibited by MLFs—making available convergent algorithms of convex programming. Finally, the book shows how MLFs are used to solve smooth mathematical programming problems, and gives the convergence rate for those dual methods based on MLFs. For mathematicians involved in discrete math and optimization, and for graduate students taking courses in complex analysis and mathematical programming, Modified Lagrangians and Monotone Maps in Optimization serves as an indispensable professional reference and graduatelevel text that goes beyond the classical Lagrange scheme, and offers diverse techniques for tackling this field. How modified Lagrangian functions improve the classical Lagrange scheme—a unique guide for working out optimization problems This volume presents the theory and applications of modified Lagrangian functions. It offers here, for the first time, a detailed analysis and numerous techniques for this fastgrowing branch of mathematical programming. Focusing on two key areas, traditional convex programming and monotone maps, the book explores a number of practical applications for MLFs and shows how MLFs are especially relevant to traditional convex programming. For mathematicians and graduate students working with optimization problem analysis, this combined text and reference
