- Nelder-Mead method
simplex algorithmfor the numerical solution of the linear programmingproblem."
The Nelder-Mead method or downhill simplex method or amoeba method is a commonly used nonlinear optimization
algorithm. It is due to John Nelder& R. Mead(1965) and is a numerical methodfor minimizing an objective functionin a many-dimensional space.
The method uses the concept of a
simplex, which is a polytopeof "N" + 1 vertices in "N" dimensions; a line segment on a line, a triangle on a plane, a tetrahedronin three-dimensional space and so forth.
The method approximately finds a locally optimal solution to a problem with "N" variables when the objective function varies smoothly. For example, a suspension bridge engineer has to choose how thick each strut, cable, and pier must be. Clearly these all link together, but it is not easy to visualize the impact of changing any specific element. The engineer can use the Nelder-Mead method to generate trial designs which are then tested on a large computer model. As each run of the simulation is expensive, it is important to make good decisions about where to look. Nelder-Mead generates a new test position by extrapolating the behavior of the objective function measured at each test point arranged as a simplex. The algorithm then chooses to replace one of these test points with the new test point and so the algorithm progresses.
The simplest step is to replace the worst point with a point reflected through the
centroidof the remaining "N" points. If this point is better than the best current point, then we can try stretching exponentially out along this line. On the other hand, if this new point isn't much better than the previous value, then we are stepping across a valley, so we shrink the simplex towards the best point.
Like all general purpose multidimensional optimization algorithms, Nelder-Mead occasionally gets stuck in a rut. The standard approach to handle this is to restart the algorithm with a new simplex starting at the current best value. This can be extended in a similar way to
simulated annealingto escape small local minima.
Many variations exist depending on the actual nature of problem being solved. The most common, perhaps, is to use a constant size small simplex that climbs local gradients to local maxima. Visualize a small triangle on an elevation map flip flopping its way up a hill to a local peak. This, however, tends to perform poorly against the method described in this article because it makes small, unnecessary steps in areas of little interest.
This method is also known as the Flexible Polyhedron Method.
One possible variation of the NM algorithm
* 1. First order according to the values at the vertices:::
* 2. Compute a reflection:
:: is the center of gravity of all points except .
::If ,::then we compute a new simplex with and by rejecting . Go to step 1.
* 3. expansion: If
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