The Nelder Mead method

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
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
• Examples of simplexes include a line segment on a
line, a triangle on a plane, a tetrahedron in threedimensional 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
1. Sort by function value: Order the vertices to
satisfy f1 < f2 < … < fn+1
2. Calculate xm = sum xi (the average of all the
points except the worst)
3. 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
4. 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
5. Contraction. If fr > fn, perform a contraction
between xm and the better of xr and xn+1.
a) 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.
b) 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
6. Shrink. Evaluate f at the n points vi = xi + S (xix1), 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
Parameter 2
Example Nelder-Mead Algorithm
Parameter 1

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