Philip Wolfe

The simplex method for quadratic programming

Abstract : A computational procedure is given for finding the minimum of a quadratic function of variables subject to linear inequality constraints. The procedure is analogous to the Simplex Method for linear programming, being based on the Barankin-Dorfman procedure for this problem.

Recent Developments in Nonlinear Programming

Publisher Summary This chapter focuses on the recent developments in nonlinear programming. Nonlinear programming has been a topic of discussion among people concerned with allocation problems about as long as linear programming has. The recent past has shown a great increase in the attention devoted to this area and some of the reasons for this new interest are readily ascertained. On the theoretical side, a large part of the purely mathematical theory of linear programming is now well in hand. The practical value of the solution of these problems is high. Almost no real problem is linear; linearity represents our compromise between reality and the limitations of our tools for dealing with it. This chapter presents a survey of the majority of techniques that have been proposed for nonlinear programming problems. Disparate as they are, they may be grouped under several broad headings in such a way that the techniques belonging to one group are similar in concept, aimed at a certain class of problem, and may have similar computational effectiveness. This chapter also discusses differential gradient methods, large-step gradient methods, simplicial methods, columnar procedures, the cutting-plane method, the process of initiating an algorithm, and computer routines and literature on direct differential gradient methods, Lagrangian differential gradient methods, the simplex-corrected gradient method, projected-gradient procedures, quadratic programming, separable programming, the decomposition procedure, and the cutting-plane method .

Nonlinear programming computations

This paper is concerned with two recently developed computational algorithms for the solution of particular classes of nonlinear programming problems.