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Reviewer: Joseph M. Lambert
The book under review is the fifth edition of Linear programming: methods and applications. The fourth edition was dated 1975; the first was published in 1958. One can see the evolution of linear programming from the early works of Dantzig in the late '40s (see, e.g., [1]) to the richness of applications of linear programming by skimming the first edition and reading the fifth edition in detail. The author claims the material is appropriate for an upper division undergraduate course or a first year graduate course in linear programming. The first 180 pages could be used for an undergraduate course, but much of the richness of the text would be inaccessible to the majority of undergraduate students. The work certainly qualifies as a text for a graduate course. Others will find the book an invaluable reference. The book contains the material one would expect in a linear programming text. There is a comprehensive review of the mathematical prerequisites. The general linear programming problem, the simplex method, the revised simplex method, duality in linear programming, parametric linear programming, sensitivity analysis, multiobjective programming, and integer programming are treated. Applications to the transportation problem, network flow problems, and others, show the power of linear programming methods. A short chapter on nonlinear programming shows the delicacy of that problem area. The exposition is clear, mathematically rigorous, and comprehensive. Examples of the mathematical techniques assist the presentation. Particular care is given to problems of computation. Computer accuracy, sensitivity, sparse data, decomposition of large systems, and the computational efficiency of the simplex method are discussed. In particular, pointers to a bibliography of over 700 items will be invaluable to practitioners and for mathematically sophisticated novices to the area of linear programming. The 1980 ellipsorial algorithm of Khachiyan [2] is reported. Pointers to the reference for that work and discussion of the efficiency of that algorithm are included. The most recent (unpublished) work of Karmarkar is not mentioned. There is little information on current large-scale linear programming packages. The reader is warned to become familiar with the idiosyncrasies of the particular packages available for use. A careful reading of some of the references will lead the reader to current information on large packages. The book is a valuable addition to the library of any practitioner of linear programming. Its choice as a text will be a personal one. It certainly should be given serious consideration as a text for both undergraduate and graduate courses.
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We present branching-on-hyperplane methods for solving mixed integer linear and mixed integer convex programs. In particular, we formulate the problem of finding a good branching hyperplane using a novel concept of adjoint lattice. We also reformulate .
Linear programming (LP) is an area of applied mathematics that is concerned with the minimization (or maximization) of a real-valued linear function on a convex polyhedron. Since the underlying theory is relatively straightforward, the history of the .
This paper studies linear programming (LP) relaxations for solving polynomial programming problems. A polynomial programming problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP) by introducing .
