Non-derivative Minimisation Methods

A simplex is a geometrical figure with M + 1 interconnected vertices, where M is the dimensionality of the energy function. For a function of two variables the simplex is thus triangular in shape. A tetrahedral simplex is used for a ftmction of three variables and so for an energy function of 3N Cartesian coordinates the simplex will have 3N -t-1 vertices if internal coordinates are used then the simplex will have 3N — 5 vertices. Each vertex corresponds to a specific set of coordinates for which an energy can be calculated. For our function/(x,y) = x + the simplex method would use a triangular simplex. 

To implement the simplex algorithm it is first necessary to generate the vertices of the initial simplex. The initial configuration of the system corresponds to just one of these vertices. The remaining points can be obtained in a variety of ways, but one simple method is to add a constant increment to each coordinate in turn. The energy of the system is calculated at the new point, giving the function value for the relevant vertex. 

S A Teukolsky and W T Vetterling 1992. Numerical Redpes m Fortran. Cambridge, Cambridge University Press.) 

Let us consider the application of the simplex method to our quadratic function,/ = 

Fig 5 5 The first jew steps of the simplex algorithm with the function x + 2y The initial simplex corresponds to the triangle 123 Point 2 has the largest value of the function and the next simplex is the triangle 134. The simplex for the third step is 145. 

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