This book is devoted to quadratic programming problems which possess the so-called symmetric duality property and can be interpreted as a special class of linear complementarity problems. It also describes applications to various economic models.
The book contains a preface, fifteen basic chapters, two supplementary chapters (16–17) with a user manual and Fortran 77 program of the Lemke complementary pivot algorithm, respectively, and an index. Each basic chapter contains exercises and a bibliography list and, as a rule, GAMS command files for implementation of some computational algorithms.
Chapter 1 describes some introduction and examples of symmetry concepts especially related to duality. Chapters 2–3 contain some basics of nonlinear optimization theory with emphasis on Karush-Kuhn-Tucker conditions, whereas Chapter 4 presents pivot methods for systems of linear equations. Chapter 5 describes the usual and symmetric duality for quadratic programming problems; the latter is due to R. W. Cottle [Quart. Appl. Math., 21, 237–243 (1963; Zbl 0127.36802)]. In this case, the primal and dual problems are symmetric in some sense and both involve primal and dual variables. At the same time, they can be written in the form of the linear complementarity problem described in Chapter 6, which also contains the Lemke-Howson complementary pivot algorithm. Chapter 7 describes the linear programming theory and simplex algorithm with application to economic production models. Chapters 8–12 contain applications of these techniques to some economic market equilibrium models with imperfect and perfect competition such as monopoly, monopsony, decision problems under risky parameters, comparative statics treated as linear/quadratic parametric programming, and general market equilibrium with affine supply/demand functions. Chapter 13 is devoted to relationships with matrix and bi-matrix games. Chapter 14 presents a so-called positive mathematical programming describing technology of incorporation of empirical data (observations) in the usual mathematical programming framework. In Chapter 15, some aspects of multiple optimal solutions in linear and quadratic programming are discussed.
The book presentation is given at the student level, furnished by many numerical illustrations, and contains significant economic applications. Hence, it can be used as a textbook.