# Mathematical Programming

• Alexander S. Belenky
Part of the Applied Optimization book series (APOP, volume 20)

## Abstract

The best known method for solving a linear programming problem (we shall consider the problem in its canonical form)
that has a finite set of feasible solutions in the form of a polyhedron
is the simplex method. This method implements a geometrically clear idea of finding vertices of M that deliver the minimum value of the function 〈c, x〉 on M [1]. The idea consists of organizing a directed enumeration (exhaustive search) of vertices of M in such a way that at every step of the enumeration process, the value of 〈c, x〉 strictly decreases. The transition from one vertex of M to another one in the course of the simplex method iteration proceeds along an edge of M [2]; this edge connects the vertices, and the corresponding computations are implemented by simple linear algebraic transformations [3]. Since the number of the vertices is finite, the simplex method leads to a point of minimum of the function 〈c, x〉 on M in a finite number of steps after leaving any vertex of M used as the initial one, and the initial vertex of M is chosen by a simple algebraic acceptance [4].

## Keywords

Programming Problem Linear Programming Problem Simplex Method Initial Problem Goal Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Alexander S. Belenky

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