Genetic Algorithm and Direct Search Toolbox

patternsearch

Find the minimum of a function using a pattern search

Syntax

• x = patternsearch(@fun, x0)
  x = patternsearch(@fun, x0, A, b)
  x = patternsearch(@fun, x0, A, b, Aeq,  beq)
  x = patternsearch(@fun, x0, A, b, Aeq,  beq, lb, ub)
  x = patternsearch(@fun, x0, A, b, Aeq,  beq, lb, ub, options)
  x = patternsearch(problem)
  [x, fval] = patternsearch(@fun, x0, ...)
  [x, fval, exitflag] = patternsearch(@fun, x0, ...)
  [x, fval, exitflag, output] = patternsearch(@fun, x0, ...)
  

Description

patternsearch finds the minimum of a function using a pattern search.

x = patternsearch(@fun, x0) solves unconstrained problems of the form

where fun is a MATLAB function that computes the values of the objective function f(x), and x0 is an initial point for the pattern search algorithm. The function patternsearch accepts the objective function as a function handle of the form @fun or as an inline function. patternsearch returns a local minimum x to the objective function. The function fun accepts a vector input and returns a scalar function value.

x = patternsearch(@fun, x0, A, b) finds a local minimum x to the function fun, subject to the linear inequality constraints represented in matrix form by

If the problem has m linear inequality constraints and n variables, then

• A is a matrix of size m-by-n.
• b is a vector of length m.

x = patternsearch(@fun, x0, A, b, Aeq, beq) finds a local minimum x to the function fun, subject to the constraints

where Aeq x = beq represents the linear equality constraints in matrix form. If the problem has r linear equality constraints and n variables, then

• Aeq is a matrix of size r-by-n.
• beq is a vector of length r.

If there are no inequality constraints, pass empty matrices, [], for A and b.

x = patternsearch(@fun, x0, A, b, Aeq, beq, lb, ub) finds a local minimum x to the function fun subject to the constraints

where represents lower and upper bounds on the variables. If the problem has n variables, lb and ub are vectors of length n. If lb or ub is empty (or not provided), it is automatically expanded to -Inf or Inf, respectively. If there are no inequality or equality constraints, pass empty matrices for A, b, Aeq and beq.

x = patternsearch(@fun, x0, A, b, Aeq, beq, lb, ub, options) finds a local minimum x to the function fun, replacing the default optimization parameters by values in the structure options. You can create options with the function psoptimset. Pass empty matrices for A, b, Aeq, beq, lb, ub, and options to use the default values.

x = patternsearch(problem) finds the minimum for problem, a structure that has the following fields:

• objective -- Objective function
• X0 -- Starting point
• Aineq -- Matrix for the inequality constraints
• Bineq -- Vector for the inequality constraints
• Aeq -- Matrix for the equality constraints
• Beq -- Vector for the equality constraints
• LB -- Lower bound for x
• UB -- Upper bound for x
• options -- Options structure created with psoptimset
• randstate -- Optional field to reset the state of rand
• randnstate -- Optional field to reset the state of randn

You can create the structure problem by exporting a problem from the Pattern Search Tool, as described in Importing and Exporting Options and Problems.

Note    problem must have all the fields as specified above.

[x, fval] = patternsearch(@fun, x0, ...) returns the value of the objective function fun at the solution x.

[x, fval, exitflag] = patternsearch(@fun, x0, ...) returns exitflag, which describes the exit condition of patternsearch. If

• exitflag > 0, patternsearch converged to a solution x.
• exitflag = 0, patternsearch reached the maximum number of function evaluations or iterations.
• exitflag < 0, patternsearch did not converge to a solution.

[x, fval, exitflag, output] = patternsearch(@fun, x0, ...) returns a structure output containing information about the search. The output structure contains the following fields:

• function -- Objective function
• problemtype -- Type of problem: unconstrained, bound constrained or linear constrained
• pollmethod -- Polling method
• searchmethod -- Search method used, if any
• iteration -- Total number of iterations
• funccount -- Total number of function evaluations
• meshsize -- Mesh size at x
• message -- Reason why the algorithm terminated

Example

Given the following constraints

the following code finds the minimum of the function, lincontest6, that is provided with the toolbox:

• A = [1 1; -1 2; 2 1];
  b = [2; 2; 3];
  lb = zeros(2,1);
  [x, fval, exitflag] = patternsearch(@lincontest6,...
  [0 0],A,b,[],[],lb)
  Optimization terminated:
  Next Mesh size (9.5367e-007)less than 'TolMesh.'
  
  x =
  
      0.6667    1.3333
  
  
  fval =
  
     -8.2222
  
  
  exitflag =
  
       1
  

See Also

psearchtool, psoptimget, psoptimset

gatool

psearchtool

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