Polling Types

Using a Complete Poll in a Generalized Pattern Search

As an example, consider the following function.

f(x1,x2)={x12+x2225for x12+x2225x12+(x29)216for x12+(x29)2160otherwise.

The following figure shows a plot of the function.

 Code for generating the figure

The global minimum of the function occurs at (0, 0), where its value is -25. However, the function also has a local minimum at (0, 9), where its value is -16.

To create a file that computes the function, copy and paste the following code into a new file in the MATLAB® Editor.

function z = poll_example(x)
if x(1)^2 + x(2)^2 <= 25
    z = x(1)^2 + x(2)^2 - 25;
elseif x(1)^2 + (x(2) - 9)^2 <= 16
    z = x(1)^2 + (x(2) - 9)^2 - 16;
else z = 0;
end

Then save the file as poll_example.m in a folder on the MATLAB path.

To run a pattern search on the function, enter the following in the Optimization app:

  • Set Solver to patternsearch.

  • Set Objective function to @poll_example.

  • Set Start point to [0 5].

  • Set Level of display to Iterative in the Display to command window options.

Click Start to run the pattern search with Complete poll set to Off, its default value. The Optimization app displays the results in the Run solver and view results pane, as shown in the following figure.

The pattern search returns the local minimum at (0, 9). At the initial point, (0, 5), the objective function value is 0. At the first iteration, the search polls the following mesh points.

f((0, 5) + (1, 0)) = f(1, 5) = 0

f((0, 5) + (0, 1)) = f(0, 6) = -7

As soon as the search polls the mesh point (0, 6), at which the objective function value is less than at the initial point, it stops polling the current mesh and sets the current point at the next iteration to (0, 6). Consequently, the search moves toward the local minimum at (0, 9) at the first iteration. You see this by looking at the first two lines of the command line display.

Iter     f-count     f(x)      MeshSize     Method
    0        1         0             1      
    1        3        -7             2     Successful Poll

Note that the pattern search performs only two evaluations of the objective function at the first iteration, increasing the total function count from 1 to 3.

Next, set Complete poll to On and click Start. The Run solver and view results pane displays the following results.

This time, the pattern search finds the global minimum at (0, 0). The difference between this run and the previous one is that with Complete poll set to On, at the first iteration the pattern search polls all four mesh points.

f((0, 5) + (1, 0)) = f(1, 5) = 0

f((0, 5) + (0, 1)) = f(0, 6) = -6

f((0, 5) + (-1, 0)) = f(-1, 5) = 0

f((0, 5) + (0, -1)) = f(0, 4) = -9

Because the last mesh point has the lowest objective function value, the pattern search selects it as the current point at the next iteration. The first two lines of the command-line display show this.

Iter     f-count     f(x)      MeshSize     Method
    0        1         0             1      
    1        5        -9             2     Successful Poll

In this case, the objective function is evaluated four times at the first iteration. As a result, the pattern search moves toward the global minimum at (0, 0).

The following figure compares the sequence of points returned when Complete poll is set to Off with the sequence when Complete poll is On.

 Code for generating the figure

Compare the Efficiency of Poll Options

This example shows how several poll options interact in terms of iterations and total function evaluations. The main results are:

  • GSS is more efficient than GPS or MADS for linearly constrained problems.

  • Whether setting CompletePoll to 'on' increases efficiency or decreases efficiency is unclear, although it affects the number of iterations.

  • Similarly, whether having a 2N poll is more or less efficient than having an Np1 poll is also unclear. The most efficient poll is GSS Positive Basis Np1 with Complete poll set to on. The least efficient is MADS Positive Basis Np1 with Complete poll set to on.

    Note:   The efficiency of an algorithm depends on the problem. GSS is efficient for linearly constrained problems. However, predicting the efficiency implications of the other poll options is difficult, as is knowing which poll type works best with other constraints.

Problem setup

The problem is the same as in Performing a Pattern Search on the Example. This linearly constrained problem uses the lincontest7 file that comes with the toolbox:

  1. Enter the following into your MATLAB workspace:

    x0 = [2 1 0 9 1 0];
    Aineq = [-8 7 3 -4 9 0];
    bineq = 7;
    Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
    beq = [84 62 65 1];
  2. Open the Optimization app by entering optimtool at the command line.

  3. Choose the patternsearch solver.

  4. Enter the problem and constraints as pictured.

  5. Ensure that the Poll method is GPS Positive basis 2N.

Generate the Results

  1. Run the optimization.

  2. Choose File > Export to Workspace.

  3. Export the results to a structure named gps2noff.

  4. Set Options > Poll > Complete poll to on.

  5. Run the optimization.

  6. Export the result to a structure named gps2non.

  7. Set Options > Poll > Poll method to GPS Positive basis Np1 and set Complete poll to off.

  8. Run the optimization.

  9. Export the result to a structure named gpsnp1off.

  10. Set Complete poll to on.

  11. Run the optimization.

  12. Export the result to a structure named gpsnp1on.

  13. Continue in a like manner to create solution structures for the other poll methods with Complete poll set on and off: gss2noff, gss2non, gssnp1off, gssnp1on, mads2noff, mads2non, madsnp1off, and madsnp1on.

Examine the Results

You have the results of 12 optimization runs. The following table shows the efficiency of the runs, measured in total function counts and in iterations. Your MADS results could differ, since MADS is a stochastic algorithm.

AlgorithmFunction CountIterations
GPS2N, complete poll off1462136
GPS2N, complete poll on139696
GPSNp1, complete poll off864118
GPSNp1, complete poll on1007104
GSS2N, complete poll off75884
GSS2N, complete poll on88974
GSSNp1, complete poll off53394
GSSNp1, complete poll on49170
MADS2N, complete poll off922162
MADS2N, complete poll on2285273
MADSNp1, complete poll off1155201
MADSNp1, complete poll on1651201

To obtain, say, the first row in the table, enter gps2noff.output.funccount and gps2noff.output.iterations. You can also examine a structure in the Variables editor by double-clicking the structure in the Workspace Browser, and then double-clicking the output structure.

The main results gleaned from the table are:

  • Setting Complete poll to on generally lowers the number of iterations for GPS and GSS, but the change in number of function evaluations is unpredictable.

  • Setting Complete poll to on does not necessarily change the number of iterations for MADS, but substantially increases the number of function evaluations.

  • The most efficient algorithm/options settings, with efficiency meaning lowest function count:

    1. GSS Positive basis Np1 with Complete poll set to on (function count 491)

    2. GSS Positive basis Np1 with Complete poll set to off (function count 533)

    3. GSS Positive basis 2N with Complete poll set to off (function count 758)

    4. GSS Positive basis 2N with Complete poll set to on (function count 889)

    The other poll methods had function counts exceeding 900.

  • For this problem, the most efficient poll is GSS Positive Basis Np1 with Complete poll set to on, although the Complete poll setting makes only a small difference. The least efficient poll is MADS Positive Basis 2N with Complete poll set to on. In this case, the Complete poll setting makes a substantial difference.

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