Linear Optimization and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-infinite Programs, Volume 45
Springer New York, Apr 13, 1983 - Language Arts & Disciplines - 197 pages
A linear optimization problem is the task of minimizing a linear real-valued function of finitely many variables subject to linear con straints; in general there may be infinitely many constraints. This book is devoted to such problems. Their mathematical properties are investi gated and algorithms for their computational solution are presented. Applications are discussed in detail. Linear optimization problems are encountered in many areas of appli cations. They have therefore been subject to mathematical analysis for a long time. We mention here only two classical topics from this area: the so-called uniform approximation of functions which was used as a mathematical tool by Chebyshev in 1853 when he set out to design a crane, and the theory of systems of linear inequalities which has already been studied by Fourier in 1823. We will not treat the historical development of the theory of linear optimization in detail. However, we point out that the decisive break through occurred in the middle of this century. It was urged on by the need to solve complicated decision problems where the optimal deployment of military and civilian resources had to be determined. The availability of electronic computers also played an important role. The principal computational scheme for the solution of linear optimization problems, the simplex algorithm, was established by Dantzig about 1950. In addi tion, the fundamental theorems on such problems were rapidly developed, based on earlier published results on the properties of systems of linear inequalities.
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APPLICATIONS OF WEAK DUALITY IN UNIFORM
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applications approximation problem assume assumptions basic set calculated CC(A Chapter column vectors computational scheme Consider constraints construct Conv convex cone convex set corresponding defined denote described determine discretized problem dual pair dual preference function dual problem duality theorem elements error function Example exchange step Exercise extended Chebyshev system feasible vectors function f Gaussian Gaussian elimination given grid Hence index set inequalities interval linear combination linear optimization problem linear program linear system linearly independent lower bound mass-points mathematical matrix Maximize Minimize c y subject n-l n-l n+l n+l nonempty nonlinear system optimal basic solution optimal solution optimal value Phase points polynomial of degree primal Proof r=l r r real numbers representation result satisfy sequence simplex algorithm Slater condition solvable solved supporting hyperplane system of equations theory tion uniform approximation uniform norm unique solution vectors a(s verify yn+l zeros