The Simplex Method

The Simplex Method is a linear fitting procedure to mathematical functions which may be applied to non-linear problems. It uses linear adjustment of the parameters until some convergence criterion is met. The term 'simplex' arises because the feasible solutions for the parameters may be represented by a polytope figure called a "simplex." The simplex for the case of a function of N rates is stored as an (N+1)xN rectangular array. Each column of the array contains N rates and represents a vertex of the polytope. In the 2 rate case the simplex is a triangle, 3 rates a tetrahedron, and 4 or more rates a not easily envisaged multidimensional polytope. The algorithm is given an initial set of rates in the simplex array and proceeds to find the function minimum by a process of reflection expansion and contraction of the simplex. The algorithm invariably converges to a minimum following a series of contractions so that the final simplex contains very similar rates in each column and each row has a sum of squares which is the same to at least 4 significant figures.

The Simplex method differs from the well-known and widely used Levenberg-Marquardt and Gauss-Newton methods in that it does not use derivatives, which confers safer convergence properties to the Simplex method since it is much less prone to finding false minima. One of the more remarkable features of the Downhill Simplex algorithm is that no divisions are required. Thus the pesky "div/0 runtime error" is bypassed.

Early History of Simplex and Kinetics

The Simplex Method for function minimization was published by Nelder and Mead¹ in 1965. It was not long before it was applied to kinetic problems.

Dauwe et. al.² used the simplex method to solve for two position decay lifetimes in 1974.

Penderson³ applied the simplex method to various pharmacokinetic problems in 1977. The pharmacokinetics were in fact various multi-exponential fits. The simplex method was compared to Levenberg-Marquardt or Gauss-Newton methods using derivatives of the function with respect to the parameters. The simplex method was demonstrated to be superior in many cases.

Kohn et. al.⁴ demonstrated the use of the simplex method to solve steady state enzymatic rate laws in 1979.