UnconstrainedOptimization_SimplexMethod

Non-linear Programming

Simplex Method
Chapter 6. (p.301) Non-Linear Programming II. Unconstrained Optimization
6.7 Simplex Method
Engineering Optimization - Theory and Practice, 4th Edition
Singiresu S. Rao\

The geometric figure formed by a set of (n+1) points in an n-dimensional space is called a simplex
There three (3) parts of simplex:
a). Reflection
Xr = (1 + a)X0 - aXh
Where:
Xh := max f(Xi), i = 1:n+1
X0 := 1/n sum(Xi), i = 1:n+1 & i =! h
b). Expansion
if f(Xr) < f(Xl), where Xl := min f(Xi), i = 1:n+1
Xe = gXr + (1 - g)X0
if f(Xe) < f(Xl); Xh = Xe (successful)
if f(Xe) > f(Xl); Xh = Xr (NOT successful)
End if
c). Contraction
if f(Xr) > f(Xi), i = 1:n+1 & i =! h
if f(Xr) < f(Xh)
Xh = Xr
Xc = bXh + (1 - b)X0
elseif f(Xr) > f(Xh)
Xc = bXh + (1 - b)X0
End if
if f(Xc) < min[f(Xh),f(Xr)], Xh = Xc
otherwise, all Xi = (Xi + Xl)/2
End if
End if\