Let P⊂ ℝdbe a convex polyhedron and f: ℝd→ℝ a linear function. One studies the computational complexity of the integral ∫pexp f(xdx. It is shown that these integrals satisfy nontrivial algebraic relations, which makes possible the construction of polynomial algorithms for certain polyhedra. Examples are given of the application of exponential integrals to the calculation of volume and nonlinear programming.
Unable to display preview. Download preview PDF.
- 1.F. Preparata and M. Sheimos, Computational Geometry: Introduction [in Russian], Moscow (1989).Google Scholar
- 2.A. I. Barvinok, “Method of Newton sums in problems of combinatorial optimization,” Discrete Matem.,2, No. 1, 3–15 (1990).Google Scholar
- 3.A. I. Barvinok, “Method of statistical sums in problems of combinatorial optimization,” Algebra Analiz,2, No. 5, 63–79 (1990).Google Scholar
- 4.A. I. Barvinok, “Problems of combinatorial optimization, statistical sums, and representations of the general linear group,” Matem. Zametki,49, No. 1, 3–11 (1991).Google Scholar
- 5.I. M. Gel'fand and A. V. Zelevinskii, “Algebraic and combinatorial aspects of the general theory of hypergeometric functions,” Funkts. Analiz Prilozhen.,20, No. 3, 17–34 (1986).Google Scholar
- 6.A. N. Varchenko, “Combinatorics and topology of position of affine hyperplanes in real space,” Funkts. Analiz Prilozhen.,21, No. 1, 11–22 (1987).Google Scholar
- 7.K. Aomoto, “On the structure of integrals of power product of linear functions,” Sci. Papers College General Education, Univ. Tokyo,27, 49–61 (1977).Google Scholar
- 8.F. W. J. Olver, Introduction to Asymptotics and Special Functions, Academic Press (1974).Google Scholar
- 9.I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1. Generalized Functions and Operations Over Them [in Russian], Moscow (1959).Google Scholar
- 10.B. V. Shabat, Introduction to Complex Analysis. Part 2. Functions of Several Variables [in Russian], Moscow (1985).Google Scholar