Journal Article10.1137/0108011
The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints
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TL;DR: The gradient projection method was originally presented to the American Mathematical Society for solving linear programming problems by Dantzig et al. as discussed by the authors, and has been applied to nonlinear programming problems as well.
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Abstract: more constraints or equations, with either a linear or nonlinear objective function. This distinction is made primarily on the basis of the difficulty of solving these two types of nonlinear problems. The first type is the less difficult of the two, and in this, Part I of the paper, it is shown how it is solved by the gradient projection method. It should be noted that since a linear objective function is a special case of a nonlinear objective function, the gradient projection method will also solve a linear programming problem. In Part II of the paper [16], the extension of the gradient projection method to the more difficult problem of nonlinear constraints and equations will be described. The basic paper on linear programming is the paper by Dantzig [5] in which the simplex method for solving the linear programming problem is presented. The nonlinear programming problem is formulated and a necessary and sufficient condition for a constrained maximum is given in terms of an equivalent saddle value problem in the paper by Kuhn and Tucker [10]. Further developments motivated by this paper, including a computational procedure, have been published recently [1]. The gradient projection method was originally presented to the American Mathematical Society
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References
A Generalized inverse for matrices
Roger Penrose
- 01 Jul 1955
TL;DR: A generalization of the inverse of a non-singular matrix is described in this paper as the unique solution of a certain set of equations, which is used here for solving linear matrix equations, and for finding an expression for the principal idempotent elements of a matrix.
The simplex method for quadratic programming
TL;DR: In this article, a computational procedure for finding the minimum of a quadratic function of variables subject to linear inequality constraints is given, analogous to the Simplex Method for linear programming, being based on the Barankin-Dorfman procedure.
The Gradient Projection Method for Nonlinear Programming. Part II. Nonlinear Constraints
TL;DR: A wind deflector is detachably mounted to the roof of a vehicle, which may be either a truck or passenger car to which a trailer is hitched, or in the case of a truck which may include a relatively high cargo compartment or camper body, whereby the wind is deflected upwardly so as to pass over the trailer, cargo compartment, camper bodies and reduce the wind resistance which such units normally present.
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