Genetic Algorithm and Direct Search Toolbox

Pattern Search Terminology

This section explains some standard terminology for pattern search, including

• Patterns
• Meshes
• Polling

Patterns

A pattern is a collection of vectors that the algorithm uses to determine which points to search at each iteration. For example, if there are two independent variables in the optimization problem, the default pattern consists of the following vectors.

v₁ = [1 0]
v₂ = [0 1]
v₃ = [-1 0]
v₄ = [0 -1]

The following figure shows these vectors.

Meshes

At each step, the pattern search algorithm searches a set of points, called a mesh, for a point that improves the objective function. The algorithm forms the mesh by

1. Multiplying the pattern vectors by a scalar, called the mesh size
2. Adding the resulting vectors to the current point -- the point with the best objective function value found at the previous step

For example, suppose that

• The current point is [1.6 3.4].
• The pattern consists of the vectors

v₁ = [1 0]
v₂ = [0 1]
v₃ = [-1 0]
v₄ = [0 -1]

• The current mesh size is 4.

The algorithm multiplies the pattern vectors by 4 and adds them to the current point to obtain the following mesh.

• [1.6 3.4] + 4*[1 0] = [5.6 3.4]
  [1.6 3.4] + 4*[0 1] = [1.6 7.4]
  [1.6 3.4] + 4*[-1 0] = [-2.4 3.4]
  [1.6 3.4] + 4*[0 -1] = [1.6 -0.6]
  

The pattern vector that produces a mesh point is called its direction.

Plotting the Objective Function Values and Mesh Sizes

Polling

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