patternsearch
finds a sequence of points, x0
, x1
, x2
,
... , that approach an optimal point. The value of the objective function
either decreases or remains the same from each point in the sequence
to the next. This section explains how pattern search works for the
function described in Optimize Using the GPS Algorithm.
To simplify the explanation, this section describes how the
generalized pattern search (GPS) works using a maximal positive basis
of 2N, with Scale set to Off
in Mesh options.
This section does not show how the patternsearch
algorithm
works with bounds or linear constraints. For bounds and linear constraints, patternsearch
modifies
poll points to be feasible, meaning to satisfy all bounds and linear
constraints.
This section does not encompass nonlinear constraints. To understand
how patternsearch
works with nonlinear constraints,
see Nonlinear Constraint Solver Algorithm.
The problem setup:
The pattern search begins at the initial point x0
that
you provide. In this example, x0 = [2.1 1.7]
.
At the first iteration, the mesh size is 1 and the GPS algorithm
adds the pattern vectors to the initial point x0 = [2.1 1.7]
to compute the following mesh points:
[1 0] + x0 = [3.1 1.7] [0 1] + x0 = [2.1 2.7] [-1 0] + x0 = [1.1 1.7] [0 -1] + x0 = [2.1 0.7]
The algorithm computes the objective function at the mesh points
in the order shown above. The following figure shows the value of ps_example
at
the initial point and mesh points.
The algorithm polls the mesh points by computing their objective
function values until it finds one whose value is smaller than 4.6347,
the value at x0
. In this case, the first such point
it finds is [1.1 1.7]
, at which
the value of the objective function is 4.5146
,
so the poll at iteration 1 is successful. The
algorithm sets the next point in the sequence equal to
x1 = [1.1 1.7]
Note
By default, the GPS pattern search algorithm stops the current
iteration as soon as it finds a mesh point whose fitness value is
smaller than that of the current point. Consequently, the algorithm
might not poll all the mesh points. You can make the algorithm poll
all the mesh points by setting Complete poll to |
After a successful poll, the algorithm multiplies the current mesh size by 2, the default value of Expansion factor in the Mesh options pane. Because the initial mesh size is 1, at the second iteration the mesh size is 2. The mesh at iteration 2 contains the following points:
2*[1 0] + x1 = [3.1 1.7] 2*[0 1] + x1 = [1.1 3.7] 2*[-1 0] + x1 = [-0.9 1.7] 2*[0 -1] + x1 = [1.1 -0.3]
The following figure shows the point x1
and
the mesh points, together with the corresponding values of ps_example
.
The algorithm polls the mesh points until it finds one whose
value is smaller than 4.5146, the value at x1
.
The first such point it finds is [-0.9 1.7]
,
at which the value of the objective function is 3.25
,
so the poll at iteration 2 is again successful. The algorithm sets
the second point in the sequence equal to
x2 = [-0.9 1.7]
Because the poll is successful, the algorithm multiplies the current mesh size by 2 to get a mesh size of 4 at the third iteration.
By the fourth iteration, the current point is
x3 = [-4.9 1.7]
and the mesh size is 8, so the mesh consists of the points
8*[1 0] + x3 = [3.1 1.7] 8*[0 1] + x3 = [-4.9 9.7] 8*[-1 0] + x3 = [-12.9 1.7] 8*[0 -1] + x3 = [-4.9 -1.3]
The following figure shows the mesh points and their objective function values.
At this iteration, none of the mesh points has a smaller objective
function value than the value at x3
, so the poll
is unsuccessful. In this case, the algorithm
does not change the current point at the next iteration. That is,
x4 = x3;
At the next iteration, the algorithm multiplies the current mesh size by 0.5, the default value of Contraction factor in the Mesh options pane, so that the mesh size at the next iteration is 4. The algorithm then polls with a smaller mesh size.
Setting the PollMethod
option to 'MADSPositiveBasis2N'
or 'MADSPositiveBasisNp1'
causes patternsearch
to
use both a different poll type and to react to polling differently
than the other polling algorithms.
A MADS poll uses newly generated pseudorandom mesh vectors at each iteration. The vectors are randomly shuffled components from the columns of a random lower-triangular matrix. The components of the matrix have integer sizes up to 1/$$\sqrt{\text{meshsize}}$$. In the poll, the mesh vectors are multiplied by the mesh size, so the poll points can be up to $$\sqrt{\text{meshsize}}$$ from the current point.
Unsuccessful polls contract the mesh by a factor of 4
,
ignoring the MeshContractionFactor
option. Similarly,
successful polls expand the mesh by a factor of 4
,
ignoring the MeshExpansionFactor
option. The maximum
mesh size is 1
, despite any setting of the MaxMeshSize
option.
In addition, when there is a successful poll, patternsearch
starts
at the successful point and polls again. This extra poll uses the
same mesh vectors, expanded by a factor of 4
while
staying below size 1
. The extra poll looks again
along the same directions that were just successful.
You can display the results of the pattern search at each iteration
by setting Level of display to Iterative
in
the Display to command window options. This enables
you to evaluate the progress of the pattern search and to make changes
to options
if necessary.
With this setting, the pattern search displays information about each iteration at the command line. The first four iterations are
Iter f-count f(x) MeshSize Method 0 1 4.63474 1 1 4 4.51464 2 Successful Poll 2 7 3.25 4 Successful Poll 3 10 -0.264905 8 Successful Poll 4 14 -0.264905 4 Refine Mesh
The entry Successful Poll
below Method
indicates
that the current iteration was successful. For example, the poll at
iteration 2 is successful. As a result, the objective function value
of the point computed at iteration 2, displayed below f(x)
,
is less than the value at iteration 1.
At iteration 4, the entry Refine Mesh
tells
you that the poll is unsuccessful. As a result, the function value
at iteration 4 remains unchanged from iteration 3.
By default, the pattern search doubles the mesh size after each successful poll and halves it after each unsuccessful poll.
The pattern search performs 60 iterations before stopping. The following plot shows the points in the sequence computed in the first 13 iterations of the pattern search.
The numbers below the points indicate the first iteration at which the algorithm finds the point. The plot only shows iteration numbers corresponding to successful polls, because the best point doesn't change after an unsuccessful poll. For example, the best point at iterations 4 and 5 is the same as at iteration 3.
At each iteration, the pattern search polls the points in the
current mesh—that is, it computes the objective function at
the mesh points to see if there is one whose function value is less
than the function value at the current point. How Pattern Search Polling Works provides an example of polling.
You can specify the pattern that defines the mesh by the Poll
method option. The default pattern, GPS Positive
basis 2N
, consists of the following 2N directions,
where N is the number of independent variables
for the objective function.
[1 0 0...0]
[0 1 0...0]
...
[0 0 0...1]
[–1 0 0...0]
[0 –1 0...0]
[0 0 0...–1].
For example, if the objective function has three independent
variables, the GPS Positive basis 2N
, consists
of the following six vectors.
[1 0 0]
[0 1 0]
[0 0 1]
[–1 0 0]
[0 –1 0]
[0
0 –1].
Alternatively, you can set Poll method to GPS
Positive basis NP1
, the pattern consisting of the following N +
1 directions.
[1 0 0...0]
[0 1 0...0]
...
[0 0 0...1]
[–1 –1
–1...–1].
For example, if objective function has three independent variables,
the GPS Positive basis Np1
, consists of the following
four vectors.
[1 0 0]
[0 1 0]
[0 0 1]
[–1 –1 –1].
A pattern search will sometimes run faster using GPS
Positive basis Np1
rather than the GPS Positive
basis 2N
as the Poll method, because
the algorithm searches fewer points at each iteration. Although not
being addressed in this example, the same is true when using the MADS
Positive basis Np1
over the MADS Positive basis
2N
, and similarly for GSS. For example, if you run a pattern
search on the example described in Linearly Constrained Problem, the algorithm
performs 1588 function evaluations with GPS Positive basis
2N
, the default Poll method, but only
877 function evaluations using GPS Positive basis Np1
.
For more detail, see Compare the Efficiency of Poll Options.
However, if the objective function has many local minima, using GPS
Positive basis 2N
as the Poll method might
avoid finding a local minimum that is not the global minimum, because
the search explores more points around the current point at each iteration.
By default, if the pattern search finds a mesh point that improves the value of the objective function, it stops the poll and sets that point as the current point for the next iteration. When this occurs, some mesh points might not get polled. Some of these unpolled points might have an objective function value that is even lower than the first one the pattern search finds.
For problems in which there are several local minima, it is
sometimes preferable to make the pattern search poll all the
mesh points at each iteration and choose the one with the best objective
function value. This enables the pattern search to explore more points
at each iteration and thereby potentially avoid a local minimum that
is not the global minimum. In the Optimization app you can make the
pattern search poll the entire mesh setting Complete poll to On
in Poll options.
At the command line, use optimoptions
to set
the UseCompletePoll
option to true
.
The criteria for stopping the pattern search algorithm are listed in the Stopping criteria section of the Optimization app:
The algorithm stops when any of the following conditions occurs:
The mesh size is less than Mesh tolerance.
The number of iterations performed by the algorithm reaches the value of Max iteration.
The total number of objective function evaluations performed by the algorithm reaches the value of Max function evaluations.
The time, in seconds, the algorithm runs until it reaches the value of Time limit.
The distance between the point found in two consecutive iterations and the mesh size are both less than X tolerance.
The change in the objective function in two consecutive iterations is less than Function tolerance and the mesh size is less than X tolerance.
Nonlinear constraint tolerance is not used as stopping criterion. It determines the feasibility with respect to nonlinear constraints.
The MADS algorithm uses an additional parameter called the poll parameter, Δ_{p}, in the mesh size stopping criterion:
$${\Delta}_{p}=\{\begin{array}{ll}N\sqrt{{\Delta}_{m}}\hfill & \text{forpositivebasis}N+1\text{poll}\hfill \\ \sqrt{{\Delta}_{m}}\hfill & \text{forpositivebasis2}N\text{poll,}\hfill \end{array}$$
where Δ_{m} is the mesh size. The MADS stopping criterion is:
Δ_{p} ≤ Mesh tolerance.
The pattern search algorithm is robust in relation to objective
function failures. This means patternsearch
tolerates
function evaluations resulting in NaN
, Inf
,
or complex values. When the objective function at the initial point x0
is
a real, finite value, patternsearch
treats poll
point failures as if the objective function values are large, and
ignores them.
For example, if all points in a poll evaluate to NaN
, patternsearch
considers
the poll unsuccessful, shrinks the mesh, and reevaluates. If even
one point in a poll evaluates to a smaller value than any seen yet, patternsearch
considers
the poll successful, and expands the mesh.