Duality in Mathematical Programming

Abstract

The theory of optimization has a long history in mathematics, The oldest problem known to me is the 27th Theorem of the Sixth Book of Euclid, However, the first real surge of interest came in the 17th century with the discovery of the calculus (and even before, with the work of Fermat), The applications of the techniques of the differential calculus and the calculus of variations were, in large part, to mathematics, physical sciences, and engineering. The last 30 years have seen the growth of Mathematical Programing, The problems which are considered in this area may be roughly classified as problems in economics, since when broadly defined, mathematical programming is concerned with the allocation of scarce resources so as to fulfill certain requirements while optimizing some objective function. As Neustadt has indicated in the opening ceremonies of this Summer School, it has had an impressive record of practical success in solving real problems. This has been due to three influences: (1) an intellectual readiness for mathematical models in economics and industrial application; (2) a succession of good algorithms, starting with the Simplex Method; (3) the simultaneous growth of large scale electronic computers.