Nelder Mead

Fminsearch uses Nelder Mead Fminsearch finds minimum of a function of several variables starting from an initial value. Unconstraint nonlinear optimization method, meaning we cannot give upper or lower bounds for parameters Downhill simplex method Global optimization method (finds global minimum)

Nelder Mead The Nelder–Mead technique is a heuristic search method that can converge to non-stationary points. But it is easy to use and will converge for a large class of problems. The Nelder–Mead technique was proposed by John Nelder & Roger Mead (1965). The method uses the concept of a simplex, which is a special polynomium type with N + 1 vertices in N dimensions. Examples of simplexes include a line segment on a line, a triangle on a plane, a tetrahedron in three-dimensional space and so forth.

The Nelder-Mead Algorithm Given n+1 vertices xi, i=1… n+1 and associated function values f(xi). Define the following coefficients R=1 (reflection) K=0.5 (contraction) E=2 (expansion) S=0.5 (shrinkage)

The Nelder-Mead Algorithm Sort by function value: Order the vertices to satisfy f1 < f2 < … < fn+1 Calculate xm = sum xi (the average of all the points except the worst) Reflection. Compute xr = xm + R(xm-xn+1) and evaluate f(xr). If f1 < fr < fn accept xr and terminate the iteration.

The Nelder-Mead Algorithm Expansion. If fr < f1 calculate xe = xm+ K (xr - xm) and evaluate f(xe). If fe < fr, accept xe; otherwise accept xr. Terminate the iteration.

The Nelder-Mead Algorithm Contraction. If fr > fn, perform a contraction between xm and the better of xr and xn+1. Outside. If fn < fr < fn+1 calculate xoc= xm+ K (xr - xm) and evaluate f(xoc). If foc< fr, accept xoc and terminate the iteration; otherwise do a shrink. Inside. If fr > fn+1 calculate xic = xm – K (xm- xn+1) and evaluate f(xic). If fic< fn+1 accept xic and terminate the iteration; otherwise do a shrink.

The Nelder-Mead Algorithm Shrink. Evaluate f at the n points vi = xi + S (xi-x1), i = 2,….,n+1. The vertices of the simplex at the next iteration are x1, v2, …, vn+1.

Standard NM moves in 2d

Example Nelder-Mead Algorithm

Example Nelder-Mead Algorithm Parameter 2 Parameter 1