*Last updated: 2023-03-16.*
# tf_quant_finance.math.optimizer.nelder_mead_minimize
Minimum of the objective function using the Nelder Mead simplex algorithm.
```python
tf_quant_finance.math.optimizer.nelder_mead_minimize(
objective_function, initial_simplex=None, initial_vertex=None, step_sizes=None,
objective_at_initial_simplex=None, objective_at_initial_vertex=None,
batch_evaluate_objective=False, func_tolerance=1e-08, position_tolerance=1e-08,
parallel_iterations=1, max_iterations=None, reflection=None, expansion=None,
contraction=None, shrinkage=None, name=None
)
```
Performs an unconstrained minimization of a (possibly non-smooth) function
using the Nelder Mead simplex method. Nelder Mead method does not support
univariate functions. Hence the dimensions of the domain must be 2 or greater.
For details of the algorithm, see
[Press, Teukolsky, Vetterling and Flannery(2007)][1].
Points in the domain of the objective function may be represented as a
`Tensor` of general shape but with rank at least 1. The algorithm proceeds
by modifying a full rank simplex in the domain. The initial simplex may
either be specified by the user or can be constructed using a single vertex
supplied by the user. In the latter case, if `v0` is the supplied vertex,
the simplex is the convex hull of the set:
```None
S = {v0} + {v0 + step_i * e_i}
```
Here `e_i` is a vector which is `1` along the `i`-th axis and zero elsewhere
and `step_i` is a characteristic length scale along the `i`-th axis. If the
step size is not supplied by the user, a unit step size is used in every axis.
Alternately, a single step size may be specified which is used for every
axis. The most flexible option is to supply a bespoke step size for every
axis.
### Usage:
The following example demonstrates the usage of the Nelder Mead minimzation
on a two dimensional problem with the minimum located at a non-differentiable
point.
```python
# The objective function
def sqrt_quadratic(x):
return tf.sqrt(tf.reduce_sum(x ** 2, axis=-1))
start = tf.constant([6.0, -21.0]) # Starting point for the search.
optim_results = tfp.optimizer.nelder_mead_minimize(
sqrt_quadratic, initial_vertex=start, func_tolerance=1e-8,
batch_evaluate_objective=True)
# Check that the search converged
assert(optim_results.converged)
# Check that the argmin is close to the actual value.
np.testing.assert_allclose(optim_results.position, np.array([0.0, 0.0]),
atol=1e-7)
# Print out the total number of function evaluations it took.
print("Function evaluations: %d" % optim_results.num_objective_evaluations)
```
### References:
[1]: William Press, Saul Teukolsky, William Vetterling and Brian Flannery.
Numerical Recipes in C++, third edition. pp. 502-507. (2007).
http://numerical.recipes/cpppages/chap0sel.pdf
[2]: Jeffrey Lagarias, James Reeds, Margaret Wright and Paul Wright.
Convergence properties of the Nelder-Mead simplex method in low dimensions,
Siam J. Optim., Vol 9, No. 1, pp. 112-147. (1998).
http://www.math.kent.edu/~reichel/courses/Opt/reading.material.2/nelder.mead.pdf
[3]: Fuchang Gao and Lixing Han. Implementing the Nelder-Mead simplex
algorithm with adaptive parameters. Computational Optimization and
Applications, Vol 51, Issue 1, pp 259-277. (2012).
https://pdfs.semanticscholar.org/15b4/c4aa7437df4d032c6ee6ce98d6030dd627be.pdf
#### Args:
* `objective_function`: A Python callable that accepts a point as a
real `Tensor` and returns a `Tensor` of real dtype containing
the value of the function at that point. The function
to be minimized. If `batch_evaluate_objective` is `True`, the callable
may be evaluated on a `Tensor` of shape `[n+1] + s ` where `n` is
the dimension of the problem and `s` is the shape of a single point
in the domain (so `n` is the size of a `Tensor` representing a
single point).
In this case, the expected return value is a `Tensor` of shape `[n+1]`.
Note that this method does not support univariate functions so the problem
dimension `n` must be strictly greater than 1.
* `initial_simplex`: (Optional) `Tensor` of real dtype. The initial simplex to
start the search. If supplied, should be a `Tensor` of shape `[n+1] + s`
where `n` is the dimension of the problem and `s` is the shape of a
single point in the domain. Each row (i.e. the `Tensor` with a given
value of the first index) is interpreted as a vertex of a simplex and
hence the rows must be affinely independent. If not supplied, an axes
aligned simplex is constructed using the `initial_vertex` and
`step_sizes`. Only one and at least one of `initial_simplex` and
`initial_vertex` must be supplied.
* `initial_vertex`: (Optional) `Tensor` of real dtype and any shape that can
be consumed by the `objective_function`. A single point in the domain that
will be used to construct an axes aligned initial simplex.
* `step_sizes`: (Optional) `Tensor` of real dtype and shape broadcasting
compatible with `initial_vertex`. Supplies the simplex scale along each
axes. Only used if `initial_simplex` is not supplied. See description
above for details on how step sizes and initial vertex are used to
construct the initial simplex.
* `objective_at_initial_simplex`: (Optional) Rank `1` `Tensor` of real dtype
of a rank `1` `Tensor`. The value of the objective function at the
initial simplex. May be supplied only if `initial_simplex` is
supplied. If not supplied, it will be computed.
* `objective_at_initial_vertex`: (Optional) Scalar `Tensor` of real dtype. The
value of the objective function at the initial vertex. May be supplied
only if the `initial_vertex` is also supplied.
* `batch_evaluate_objective`: (Optional) Python `bool`. If True, the objective
function will be evaluated on all the vertices of the simplex packed
into a single tensor. If False, the objective will be mapped across each
vertex separately. Evaluating the objective function in a batch allows
use of vectorization and should be preferred if the objective function
allows it.
* `func_tolerance`: (Optional) Scalar `Tensor` of real dtype. The algorithm
stops if the absolute difference between the largest and the smallest
function value on the vertices of the simplex is below this number.
* `position_tolerance`: (Optional) Scalar `Tensor` of real dtype. The
algorithm stops if the largest absolute difference between the
coordinates of the vertices is below this threshold.
* `parallel_iterations`: (Optional) Positive integer. The number of iterations
allowed to run in parallel.
* `max_iterations`: (Optional) Scalar positive `Tensor` of dtype `int32`.
The maximum number of iterations allowed. If `None` then no limit is
applied.
* `reflection`: (Optional) Positive Scalar `Tensor` of same dtype as
`initial_vertex`. This parameter controls the scaling of the reflected
vertex. See, [Press et al(2007)][1] for details. If not specified,
uses the dimension dependent prescription of [Gao and Han(2012)][3].
* `expansion`: (Optional) Positive Scalar `Tensor` of same dtype as
`initial_vertex`. Should be greater than `1` and `reflection`. This
parameter controls the expanded scaling of a reflected vertex.
See, [Press et al(2007)][1] for details. If not specified, uses the
dimension dependent prescription of [Gao and Han(2012)][3].
* `contraction`: (Optional) Positive scalar `Tensor` of same dtype as
`initial_vertex`. Must be between `0` and `1`. This parameter controls
the contraction of the reflected vertex when the objective function at
the reflected point fails to show sufficient decrease.
See, [Press et al(2007)][1] for more details. If not specified, uses
the dimension dependent prescription of [Gao and Han(2012][3].
* `shrinkage`: (Optional) Positive scalar `Tensor` of same dtype as
`initial_vertex`. Must be between `0` and `1`. This parameter is the scale
by which the simplex is shrunk around the best point when the other
steps fail to produce improvements.
See, [Press et al(2007)][1] for more details. If not specified, uses
the dimension dependent prescription of [Gao and Han(2012][3].
* `name`: (Optional) Python str. The name prefixed to the ops created by this
function. If not supplied, the default name 'minimize' is used.
#### Returns:
* `optimizer_results`: A namedtuple containing the following items:
converged: Scalar boolean tensor indicating whether the minimum was
found within tolerance.
num_objective_evaluations: The total number of objective
evaluations performed.
position: A `Tensor` containing the last argument value found
during the search. If the search converged, then
this value is the argmin of the objective function.
objective_value: A tensor containing the value of the objective
function at the `position`. If the search
converged, then this is the (local) minimum of
the objective function.
final_simplex: The last simplex constructed before stopping.
final_objective_values: The objective function evaluated at the
vertices of the final simplex.
initial_simplex: The starting simplex.
initial_objective_values: The objective function evaluated at the
vertices of the initial simplex.
num_iterations: The number of iterations of the main algorithm body.
#### Raises:
* `ValueError`: If any of the following conditions hold
1. If none or more than one of `initial_simplex` and `initial_vertex` are
supplied.
2. If `initial_simplex` and `step_sizes` are both specified.