Linear-fractional programming. Theory, methods, applications and software.
Linear-fractional programming consists in maximizing or minimizing the quotient of two affine functions subject to linear constraints. This branch of mathematical programming dates back to the pioneering work of \textit{B. Martós} [Publ. Math. Inst. Hung. Acad. Sci., Ser. B 5, 383--406 (1960; Zbl 0099.15101)], who called it hyperbolic programming. Although several books on general (nonlinear) fractional programming are available, this is the first monograph entirely devoted to linear-fractional problems. As the title indicates, it deals with both the theoretical and the algorithmic aspects of the subject, including the description of a program package for Windows. The chapter headings are: 1. Introduction; 2. Basic linear algebra; 3. Introduction to LFP; 4. The simplex method; 5. Duality theory; 6. Sensitivity analysis; 7. Interconnection between LFP and LP; 8. Integer linear-fractional programming; 9. Special LFP problems; 10. Advanced methods and algorithms in LFP; 11. Advanced topics in LFP; 12. Computational aspects; 13. The WinGULF package. The rather long, though non-exhaustive, list of references contains 191 items. The theoretical issues discussed in the book include several approaches to duality theory, among which the author chooses the one due to \textit{E. G. Golstein} [Sov. Math., Dokl. 8, 212--216 (1967); translation from Dokl. Akad. Nauk SSSR 172, 1007--1010 (1967; Zbl 0189.19701)], which is constructed using a fractional Lagrangian, for a more detailed study. Concerning algorithms, the book concentrates on the simplex method and its variants, whereas only a short section is devoted to mention interior point methods. Classical applications of linear-fractional programming to problems arising in economics and operations research are briefly discussed. The last chapter describes WinGULF, a commercial package for solving linear-fractional problems in continuous or integer variables; a free student edition of this software is downloadable from the author's homepage.
scientific article; zbMATH DE number 2010207