Grid Restrained Nelder-Mead, a multivariate rootfinder.

cl-grnm

# cl-grnm

Common-Lisp implementations of the grid-restrained and traditional Nelder-Mead algorithms


## Authorship

This package was originally written by Mario S. Mommer (2006), and this fork is maintained by Rigetti Computing.


## Introduction

These common lisp sources contain two variants of the Nelder-Mead algorithm. The original algorithm [1] and a provably convergent, reliable variant by A. Bürmen et al [4], called the "Grid Restrained Nelder Mead Algorithm" (GRNMA).

It should be mentioned that other, provably convergent variant exist [2,3], which aren't included here. The only reasons are lack of time, and the fact that the implemented variant does not require a simple descent condition, putting it closer to the original.

Other than that, and based on the article [4], the performance of these methods seems to be about equal in terms of number of function evaluations. As a side effect of the additional reliability, both tend to be a lot more efficient than the original algorithm even when it does not fail. In particular when the number of dimensions increases. As a test, one might try the GRNM,

```
(grnm-optimize #'standard-quadratic (make-array 30 :initial-element 1.0d0) :verbose t)
```

and compare with the original Nelder-Mead,

```
(nm-optimize #'standard-quadratic (make-array 30 :initial-element 1.0d0) :verbose t)
```

and observe the difference in number of function evaluations (the last value returned).

(This exercise also serves to illustrate the overall deficiencies of direct search algorithms when applied to higher dimensional problems.)

This software is provided under the MIT license; see LICENSE.txt for details.


## Usage

This implementation of the grid restrained Nelder-Mead algorithm expects at least two parameters: the objective function, and an initial guess. It returns four values:

* the minimizer,
* the minimum,
* the last simplex,
* and the number of function evaluations.

The objective function should accept a double-float array as its argument.

The initial guess will usually be an array of double-float numbers (but instead an object of the class `NM-SIMPLEX` may also be provided; see below). For example,

```
(grnm-optimize #'rosenbrock #(40.0d0 40.0d0))
```

finds the minimum of the Rosenbrock function, and

```
(grnm-optimize #'standard-quadratic (make-array 30 :initial-element 1.0d0))
```

finds the minimum of the standard quadratic function in 30 dimensions.

A few keyword arguments can customize the behavior. These are

* `:verbose`  (default: `NIL`)
* `:converged-p` (default: `burmen-et-al-convergence-test`)
* `:max-function-calls` (default: `NIL`; => as many as needed)

### `:verbose` (default: `NIL`)

Pass T here if you want to see some progress report. The amount of output can be controlled by setting *verbose-level* to 1 or 2. The difference is that with 1 (the default) only the best value of the simplex is shown, while with 2 the whole simplex is printed on each iteration.

### `:converged-p` (default: `burmen-et-al-convergence-test`)

The `burmen-et-al-convergence-test` is, as the name suggests, the convergence test used in the article by Bürmen et al. It accepts a few parameters: `tol-x`, `tol-f` and `rel`; please see the article for further details.

Another convergence criterion that can be given is `(pp-volume-test <tol>)`, which returns true once the parallelogram(!) spanned by the vertices of the simplex has a volume lower than `<tol>` to the power of N, where N is the dimension of the problem. This is a rather expensive test, as it involves computing a QR decomposition of an N by N matrix on each call.

If you are not in the mood of taking prisoners, you might as well pass (constantly NIL) as the convergence criterion. This has as a consequence that the iteration continues until the simplex collapses, which in floating-point arithmetic happens in finite time. The grid restrained Nelder-Mead algorithm should have converged by then.

### `:max-function-calls` (default: `NIL`)

Maximum number of objective function evaluations. After that many function evaluations, this implementation of the algorithm will declare convergence to have occurred.

The actual number of function calls might be slightly larger (at most by N), as the relevant condition is only checked in certain situations.

## Utilities/Misc

### `NM-optimize`

Apart from the grid-restrained Nelder Mead algorithm, the traditional variant is also provided. The corresponding function is named `NM-optimize`, and its usage is the same as for `GRNM-optimize`.

The only difference is that :max-function-calls has a default value of 100000. Otherwise the algorithm might well iterate forever.

### `initial-simplex <initial guess> :displace <displacement>`

Constructs an initial simplex with the double-float array `<initial guess>` as on of its corners.

The displacement can be a an array of double floats or a number. In the first case, the additional vertices of the simplex are build by adding to each component of the `<initial guess>` the corresponding component in `<displacement>`. If it is a number, then the additional vertices are build by adding `<displacement>` to each component of the `<initial guess>`.

## Tips & Tricks

* One can try to solve constrained optimization problems by returning `MOST-POSITIVE-DOUBLE-FLOAT` whenever the objective function is called with an argument that violates the constraints. Mathematically, this falls out of the theory, but it works most of the time.
* If your objective function is noisy or not smooth (for instance, if its first derivatives are not continuous) it is a good idea to restart the algorithm. Remember that convergence is only guaranteed if the objective function is at least C^1.

## References

[1] J.A. Nelder and R. Mead, "A simplex method for function minimization," The Computer Journal, vol. 7, pp. 308-313, 1965.

[2] P. Tseng, "Fortified-descent simplicial search method: A general approach," SIAM Journal on Optimization, vol. 10, pp. 269-288, 1999.

[3] C.J. Price, I.D. Coope, and D. Byatt, "A convergent variant of the Nelder-Mead algorithm," Journal of Optimization Theory and Applications, vol. 113, pp. 5-19, 2002.

[4] A. Bürmen, J. Puhan and T. Tuma, "Grid Restrained Nelder-Mead Algorithm", Computational Optimization and Applications, vol. 34, no. 3, pp. 359 - 375, 2006.



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