Dantzig’s simplex method

When the criterion optimized is a linear function of the operating variables, the feasibility problem is said to be one in linear programming. Being the simplest possible feasibility problem, it was the first one studied, and the publication in 1951 of G. Dantzig s simplex method for solving linear-programming problems (D2) marked the beginning of contemporary research in optimization theory. Section IV,C is devoted to this important technique. 

In the previous example, the technique used to reduce the artificial variables to zero was in fact Dantzig s simplex method. The linear function optimized was the simple sum of the artificial variables. Any linear function may be optimized in the same manner. The process must start with a basic solution feasible with the constraints, the function to be optimized expressed only in terms of the variables not in the starting basis. From these expressions it is decided what nonbasic variable should be brought into the basis and what basic variable should be forced out. The process is iterated until no further improvement is possible. 

The Nelder-Mead simplex algorithm was published already on 1965, and it has become a classic (Nelder Mead, 1965). Several variants and applications of it have been published since then. It is often also called the flexible polyhedron method. It should be noted that it has nothing to do with the so-called Dantzig s simplex method used in linear programming. It can be used both in mathematical and empirical optimization. 

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