Presentation on theme: "Nelder Mead. Fminsearch uses Nelder Mead Fminsearch finds minimum of a function of several variables starting from an initial value. Unconstraint nonlinear."— Presentation transcript:
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 x i, i=1… n+1 and associated function values f(x i ). 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 f 1 < f 2 < … < f n+1 2.Calculate x m = sum x i (the average of all the points except the worst) 3.Reflection. Compute x r = x m + R(x m -x n+1 ) and evaluate f(x r ). If f 1 < f r < f n accept x r and terminate the iteration.
The Nelder-Mead Algorithm 4.Expansion. If f r < f 1 calculate x e = x m + K (x r - x m ) and evaluate f(x e ). If f e < f r, accept x e ; otherwise accept x r. Terminate the iteration.
The Nelder-Mead Algorithm 5.Contraction. If f r > f n, perform a contraction between x m and the better of x r and x n+1. a)Outside. If f n < f r < f n+1 calculate x oc = x m + K (x r - x m ) and evaluate f(x oc ). If f oc < f r, accept x oc and terminate the iteration; otherwise do a shrink. b)Inside. If f r > f n+1 calculate x ic = x m – K (x m - x n+1 ) and evaluate f(x ic ). If f ic < f n+1 accept x ic and terminate the iteration; otherwise do a shrink.
The Nelder-Mead Algorithm 6.Shrink. Evaluate f at the n points v i = x i + S (x i - x 1 ), i = 2,….,n+1. The vertices of the simplex at the next iteration are x 1, v 2, …, v n+1.