Modified Lagrangians and Monotone Maps in Optimization (Wiley Series in Discrete Mathematics and Optimization (Book 30))

This 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, de-composition, 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 mono-tone maps are treated broadly, in line with their growing importance in optimization and mathematical economics. Two chapters on monotone maps cover point-to-set maps, propose modifications that would achieve a point-to-point 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 graduate-level 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 fast-growing 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

• Describes the benefits of MLFs in applications such as numerical algorithms for the general convex programming problem, decomposition, economic modeling, nonconvex local constrained optimization, and more
• Uses the concepts of MLFs and WMLFs (weak modified Lagrangian functions) to develop a theory of duality, and illustrates the analysis with an elementary economic model
• Covers convex programming methods that are based on the iterative solution of dual problems generated by MLFs, showing how the proper choice of an MLF can guarantee the smoothness of the results
• Discusses monotone maps in much more detail than has been done to date in the professional literature, and explains how to use new MLF constructions to solve equations associated with monotone maps
• Considers convergence properties in MLFs, and how they relate to the saddle gradient method and to problem solving in convex programming
• Shows how to solve smooth mathematical programming problems, and includes results that relate to the convergence rate of the dual methods based on MLFs

E.G. Golshtein, PhD, is the Head of the Laboratory for Optimization Theory and Computational Method of the Central Economics and Mathematics Institute at the Russian Academy of Sciences and Professor of Mathematics on the economic faculty, Moscow State University. He is also a member of the editorial boards of the journals Ekonomica i Matematicheskie Metody (Moscow), Optimization (Berlin), and Optimization: Methods and Software and has over 140 publications to his credit.

N.V. Tretyakov, PhD, is currently the Head of the LAN and DTP Laboratory at the Central Economics and Mathematics Institute (CEMI). He received his doctoral degree in mathematics from CEMI's Russian Academy of Sciences, where he has been on the research staff since 1969. Dr. Tretyakov's fields of interest are mathematical programming and computer networking.