nOnYJHIPHbIE JlEKl llil1 no MATEMATMKE A C COJIO,nOBHHKOB CllCTEMbI JII1HE:HHhIX HEPABEHCTB H3JlATEJIbCTBO «HAYKA» MOCKBA - LIITLE MATHEMATICS LIBRARY A S Solodovnikov SYSTEMS OF LINEAR INEQUALITIES Translated from the Russian by Vladimir Shokurov MIR PUBLISHERS MOSCOW First published 1979 Revised from the 1977 Russian edition Ha auesuucxou â â Jl3blKe 113,UaTeJIhCTBO ôHaYKaằ, 1977 English translation, M ir Publishers, 1979 CONTENTS ~ ~~ Some Facts from Analytic Geometry ~ Visualization of Systems of Linear Inequalities In Two or Three " Unknowns 17 I - The Convex Hull of a System of Points A Convex Polyhedral Cone 22 25 The Feasible Region of a System of Linear Inequalities in Two Unknowns 31 The Feasible Region of a System in Three Unknowns 44 Systems of Linear Inequalities in Any Number of Unknowns 52 The Solution of a System of Linear Inequalities 'by Successive Reduction of the Number of Unknowns 57 Incompatible Systems 64 10 A· Homogeneous System of Linear Inequalities The Fundamental Set of Solutions 69 11 The Solution of a Nonhomogeneous System of Inequalities 81 12 A Linear Programming Problem 84 13 The Simplex Method 91 14 The Duality Theorem in Linear Programming 101 1.5 Transportation Problem 107 Preface First-degree or, to use the generally accepted term, linear inequalities are inequalities of the form ax + by + c~ (for simplicity we have written an inequality in two unknowns x and y).The theory of systems of linear inequalities is a small but most fascinating branch of mathematics Interest in it is to a considerable extent due to the beauty of geometrical content, for in geometrical terms giving a system of linear inequalities in two or three unknowns means giving a convex polygonal region in the plane or a convex polyhedral solid in space, respectively For example, the study of convex polyhedra, a part of geometry as old as the hills, turns thereby into one of the chapters of the theory of systems of linear inequalities, This theory has also some branches which are near the algebraist's heart; for example, they include a remarkable analogy between the properties of linear inequalities and those of systems of linear equations (everything connected with linear equations has been studied for a long time and in much detail) Until recently one might think that linear inequalities would forever remain an object of purely mathematical work The situation has changed radically since the mid 40s of this century when there arose a new area of applied mathematics -linear programmingwith important applications in the economy and engineering Linear programming is in the end nothing but a part (though a very important one) of the theory of systems of linear inequalities It is exactly the aim of this small book to acquaint the reader with the various aspects of the theory of systems of linear inequalities, viz with the geometrical aspect of the matter and some of the methods for solving systems connected with that aspect, with certain purely algebraic properties of the systems, and with questions of linear programming Reading the book will not require any knowledge beyond the school course in mathematics A few words are in order about the history of the questions to be elucidated in this book Although by its subject-matter the theory of linear inequalities' should, one would think, belong to the most basic and elementary parts of mathematics, until recently it was studied relatively little From the last years of the last century works began occasionally to appear which elucidated some properties of systems of linear inequalities In this connection one can mention the names of such mathe- maticians as H Minkowski (one of the greatest geometers of the end of the last and the beginning of this century especially- well known for his works on convex sets and as the creator of "Minkowskian geometry"), G F Voronoi (one of the fathers of the "Petersburg school of number theory"), A Haar (a Hungarian mathematician who won recognition for his works on "group integration"), HiWeyl (one of the most outstanding mathematicians of the first half of this century; one can read about his life and work in the pamphlet "Herman Weyl" by I M Yaglom, Moscow, "Znanie", 1967) Some of the results obtained by them are to some extent or other reflected in the present book (though without mentioning the authors' names) It was not until the 1940s or 1950s,when the rapid growth of applied disciplines (linear, convex and other modifications of "mathematical programming", the so-called "theory of games", etc.) made an advanced and systematic study of linear inequalities a necessity, that a really intensive development of the theory of systems of linear inequalities began At present a complete list of books and papers on inequalities would probably contain hundreds of titles Some Facts from Analytic Geometry r Operations on points Consider a plane with a rectangular coordinate system The fact that a point M has coordinates x and y In this system is written down as follows: M = (x, y) or simply M(x, y) The presence of a coordinate system allows one to perform some operations on the points of the plane, namely the operation of addition of points and the operation of multiplication of a point by a number The addition of points is defined in the following way: if M = (Xb Yl) and M = (Xl, Y2), then M 1+M2 = (Xl + X2, Yl+Y2) Thus the addition of points is reduced to the addition of their similar coordinates The visualization of this operation is very simple (Fig 1); the point M + M is the fourth vertex of the parallelogram constructed on the segments OM and OM as its sides (0 is the origin of coordinates) M 1, 0, M are the three remaining vertices of the parallelogram The same can be said in another way: the point M + M is obtained by translating the point M in the direction of the segment OM lover a distance equal to the length of the segment The multiplication of the point M(x,y) by an arbitrary number k is carried out according to the following rule: kM = ikx; ky) The visualization of this operation is still simpler than that of the addition; for k > the point M' = kM lies on the ray OM, with Fig OM' = k x OM; for k < the point M' lies on the extension of the ray OM beyond the point 0, with OM' = Ikl x OM (Fig 2) The derivation of the above visualization of both operations will provide a good exercise for the reader* !I !J k/1 (k>O) # 71 o :c :JJ '/(/1 (k