patternsearch

Find minimum of function using pattern search

Syntax

• ``x = patternsearch(fun,x0)``
example
• ``x = patternsearch(fun,x0,A,b)``
example
• ``x = patternsearch(fun,x0,A,b,Aeq,beq)``
• ``x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub)``
example
• ``x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)``
example
• ``x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)``
example
• ``x = patternsearch(problem)``
• ``````[x,fval] = patternsearch(___)``````
example
• ``````[x,fval,exitflag,output] = patternsearch(___)``````
example

Description

example

````x = patternsearch(fun,x0)` finds a local minimum, `x`, to the function handle `fun` that computes the values of the objective function. `x0` is a real vector specifying an initial point for the pattern search algorithm.Note:   Passing Extra Parameters explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary.```

example

````x = patternsearch(fun,x0,A,b)` minimizes `fun` subject to the linear inequalities `A*x` ≤ `b`. See Linear Inequality Constraints.```
````x = patternsearch(fun,x0,A,b,Aeq,beq)` minimizes `fun` subject to the linear equalities `Aeq*x` = `beq` and `A*x` ≤ `b`. If no linear inequalities exist, set `A = []` and `b = []`.```

example

````x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub)` defines a set of lower and upper bounds on the design variables in `x`, so that the solution is always in the range `lb `≤` x `≤` ub`. If no linear equalities exist, set `Aeq = []` and ```beq = []```. If `x(i)` has no lower bound, set ```lb(i) = -Inf```. If `x(i)` has no upper bound, set ```ub(i) = Inf```.```

example

````x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)` subjects the minimization to the nonlinear inequalities `c(x)` or equalities `ceq(x)` defined in `nonlcon`. `patternsearch` optimizes `fun` such that `c(x) ≤ 0` and `ceq(x) = 0`. If no bounds exist, set ```lb = []```, `ub = []`, or both.```

example

````x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)` minimizes `fun` with the optimization options specified in `options`. Use `psoptimset` to set these options. If there are no nonlinear inequality or equality constraints, set ```nonlcon = []```.```
````x = patternsearch(problem)` finds the minimum for `problem`, where `problem` is a structure described in Input Arguments. Create the `problem` structure by exporting a problem from Optimization app, as described in Exporting Your Work.```

example

``````[x,fval] = patternsearch(___)```, for any syntax, returns the value of the objective function `fun` at the solution `x`.```

example

``````[x,fval,exitflag,output] = patternsearch(___)``` additionally returns `exitflag`, a value that describes the exit condition of `patternsearch`, and a structure `output` with information about the optimization process.```

Examples

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Unconstrained Pattern Search Minimization

Minimize an unconstrained problem using the `patternsearch` solver.

Create the following two-variable objective function. On your MATLAB® path, save the following code to a file named `psobj.m`.

```function y = psobj(x) y = exp(-x(1)^2-x(2)^2)*(1+5*x(1) + 6*x(2) + 12*x(1)*cos(x(2)));```

Set the objective function to `@psobj`.

`fun = @psobj;`

Find the minimum, starting at the point `[0,0]`.

```x0 = [0,0]; x = patternsearch(fun,x0)```
```Optimization terminated: mesh size less than options.TolMesh. x = -0.7037 -0.1860```

Pattern Search with a Linear Inequality Constraint

Minimize a function subject to some linear inequality constraints.

Create the following two-variable objective function. On your MATLAB path, save the following code to a file named `psobj.m`.

```function y = psobj(x) y = exp(-x(1)^2-x(2)^2)*(1+5*x(1) + 6*x(2) + 12*x(1)*cos(x(2)));```

Set the objective function to `@psobj`.

`fun = @psobj;`

Set the two linear inequality constraints.

```A = [-3,-2; -4,-7]; b = [-1;-8];```

Find the minimum, starting at the point `[0.5,-0.5]`.

```x0 = [0.5,-0.5]; x = patternsearch(fun,x0,A,b)```
```Optimization terminated: mesh size less than options.TolMesh. x = 5.2824 -1.8758```

Pattern Search with Bounds

Find the minimum of a function that has only bound constraints.

Create the following two-variable objective function. On your MATLAB path, save the following code to a file named `psobj.m`.

```function y = psobj(x) y = exp(-x(1)^2-x(2)^2)*(1+5*x(1) + 6*x(2) + 12*x(1)*cos(x(2)));```

Set the objective function to `@psobj`.

`fun = @psobj;`

Find the minimum when 0 ≤ x(1) ≤ ∞ and –∞ ≤ x(2) ≤ –3.

```lb = [0,-Inf]; ub = [Inf,-3]; A = []; b = []; Aeq = []; beq = [];```

Find the minimum, starting at the point `[1,-5]`.

```x0 = [1,-5]; x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub)```
```Optimization terminated: mesh size less than options.TolMesh. x = 0.1880 -3.0000```

Pattern Search with Nonlinear Constraints

Find the minimum of a function subject to a nonlinear inequality constraint.

Create the following two-variable objective function. On your MATLAB path, save the following code to a file named `psobj.m`.

```function y = psobj(x) y = exp(-x(1)^2-x(2)^2)*(1+5*x(1) + 6*x(2) + 12*x(1)*cos(x(2)));```

Set the objective function to `@psobj`.

`fun = @psobj;`

Create the nonlinear constraint

$\frac{xy}{2}+{\left(x+2\right)}^{2}+\frac{{\left(y-2\right)}^{2}}{2}\le 2.$

To do so, on your MATLAB path, save the following code to a file named `ellipsetilt.m`.

```function [c,ceq] = ellipsetilt(x) ceq = []; c = x(1)*x(2)/2 + (x(1)+2)^2 + (x(2)-2)^2/2 - 2;```

Start `patternsearch` from the initial point `[-2,-2]`.

```x0 = [-2,-2]; A = []; b = []; Aeq = []; beq = []; lb = []; ub = []; nonlcon = @ellipsetilt; x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)```
```Optimization terminated: mesh size less than options.TolMesh and constraint violation is less than options.TolCon. x = -1.5144 0.0874```

Pattern Search with Nondefault Options

Set options to observe the progress of the `patternsearch` solution process.

Create the following two-variable objective function. On your MATLAB path, save the following code to a file named `psobj.m`.

```function y = psobj(x) y = exp(-x(1)^2-x(2)^2)*(1+5*x(1) + 6*x(2) + 12*x(1)*cos(x(2)));```

Set the objective function to `@psobj`.

`fun = @psobj;`

Set `options` to give iterative display and to plot the objective function at each iteration.

`options = psoptimset('Display','iter','PlotFcns',@psplotbestf);`

Find the unconstrained minimum of the objective starting from the point `[0,0]`.

```x0 = [0,0]; A = []; b = []; Aeq = []; beq = []; lb = []; ub = []; nonlcon = []; x = patternsearch(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)```

```Iter f-count f(x) MeshSize Method 0 1 1 1 1 4 -5.88607 2 Successful Poll 2 8 -5.88607 1 Refine Mesh 3 12 -5.88607 0.5 Refine Mesh 4 16 -5.88607 0.25 Refine Mesh (output trimmed) 63 218 -7.02545 1.907e-06 Refine Mesh 64 221 -7.02545 3.815e-06 Successful Poll 65 225 -7.02545 1.907e-06 Refine Mesh 66 229 -7.02545 9.537e-07 Refine Mesh Optimization terminated: mesh size less than options.TolMesh. x = -0.7037 -0.1860```

Obtain Function Value And Minimizing Point

Find a minimum value of a function and report both the location and value of the minimum.

Create the following two-variable objective function. On your MATLAB path, save the following code to a file named `psobj.m`.

```function y = psobj(x) y = exp(-x(1)^2-x(2)^2)*(1+5*x(1) + 6*x(2) + 12*x(1)*cos(x(2)));```

Set the objective function to `@psobj`.

`fun = @psobj;`

Find the unconstrained minimum of the objective, starting from the point `[0,0]`. Return both the location of the minimum, `x`, and the value of `fun(x)`.

```x0 = [0,0]; [x,fval] = patternsearch(fun,x0)```
```Optimization terminated: mesh size less than options.TolMesh. x = -0.7037 -0.1860 fval = -7.0254```

Obtain All Outputs

To examine the `patternsearch` solution process, obtain all outputs.

Create the following two-variable objective function. On your MATLAB path, save the following code to a file named `psobj.m`.

```function y = psobj(x) y = exp(-x(1)^2-x(2)^2)*(1+5*x(1) + 6*x(2) + 12*x(1)*cos(x(2)));```

Set the objective function to `@psobj`.

`fun = @psobj;`

Find the unconstrained minimum of the objective, starting from the point `[0,0]`. Return the solution, `x`, the objective function value at the solution, `fun(x)`, the exit flag, and the output structure.

```x0 = [0,0]; [x,fval,exitflag,output] = patternsearch(fun,x0)```
```Optimization terminated: mesh size less than options.TolMesh. x = -0.7037 -0.1860 fval = -7.0254 exitflag = 1 output = function: @psobj problemtype: 'unconstrained' pollmethod: 'gpspositivebasis2n' maxconstraint: [] searchmethod: [] iterations: 66 funccount: 229 meshsize: 9.5367e-07 rngstate: [1x1 struct] message: 'Optimization terminated: mesh size less than options.TolMesh.'```

The `exitflag` is `1`, indicating convergence to a local minimum.

The `output` structure includes information such as how many iterations `patternsearch` took, and how many function evaluations. Compare this output structure with the results from Pattern Search with Nondefault Options. In that example, you obtain some of this information, but did not obtain, for example, the number of function evaluations.

Input Arguments

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`fun` — Function to be minimizedfunction handle | function name

Function to be minimized, specified as a function handle or function name. The `fun` function accepts a vector `x` and returns a real scalar `f`, which is the objective function evaluated at `x`.

You can specify `fun` as a function handle for a file

`x = patternsearch(@myfun,x0)`

Here, `myfun` is a MATLAB function such as

```function f = myfun(x) f = ... % Compute function value at x```

`fun` can also be a function handle for an anonymous function

`x = patternsearch(@(x)norm(x)^2,x0,A,b);`

Example: `fun = @(x)sin(x(1))*cos(x(2))`

Data Types: `char` | `function_handle`

`x0` — Initial pointreal vector

Initial point, specified as a real vector. `patternsearch` uses the number of elements in `x0` to determine the number of variables that `fun` accepts.

Example: `x0 = [1,2,3,4]`

Data Types: `double`

`A` — Linear inequality constraintsreal matrix

Linear inequality constraints, specified as a real matrix. `A` is an `M`-by-`N` matrix, where `M` is the number of inequalities, and `N` is the number of variables (number of elements in `x0`).

`A` encodes the `M` linear inequalities

`A*x <= b`,

where `x` is the column vector of `N` variables `x(:)`, and `b` is a column vector with `M` elements.

For example, to specify

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,

give these constraints:

```A = [1,2;3,4;5,6]; b = [10;20;30];```

Example: To specify that the control variables sum to 1 or less, give the constraints `A = ones(1,N)` and ```b = 1```

Data Types: `double`

`b` — Linear inequality constraintsreal vector

Linear inequality constraints, specified as a real vector. `b` is an `M`-element vector related to the `A` matrix. If you pass `b` as a row vector, solvers internally convert `b` to the column vector `b(:)`.

`b` encodes the `M` linear inequalities

`A*x <= b`,

where `x` is the column vector of `N` variables `x(:)`, and `A` is a matrix of size `M`-by-`N`.

For example, to specify

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,

give these constraints:

```A = [1,2;3,4;5,6]; b = [10;20;30];```

Example: To specify that the control variables sum to 1 or less, give the constraints `A = ones(1,N)` and ```b = 1```

Data Types: `double`

`Aeq` — Linear equality constraintsreal matrix

Linear equality constraints, specified as a real matrix. `Aeq` is an `Me`-by-`N` matrix, where `Me` is the number of equalities, and `N` is the number of variables (number of elements in `x0`).

`Aeq` encodes the `Me` linear equalities

`Aeq*x = beq`,

where `x` is the column vector of `N` variables `x(:)`, and `beq` is a column vector with `Me` elements.

For example, to specify

x1 + 2x2 + 3x3 ≤ 10
2x1 + 4x2 + x3 ≤ 20,

give these constraints:

```A = [1,2,3;2,4,1]; b = [10;20];```

Example: To specify that the control variables sum to 1, give the constraints `Aeq = ones(1,N)` and ```beq = 1```

Data Types: `double`

`beq` — Linear equality constraintsreal vector

Linear equality constraints, specified as a real vector. `beq` is an `Me`-element vector related to the `Aeq` matrix. If you pass `beq` as a row vector, solvers internally convert `beq` to the column vector `beq(:)`.

`beq` encodes the `Me` linear equalities

`Aeq*x = beq`,

where `x` is the column vector of `N` variables `x(:)`, and `Aeq` is a matrix of size `Meq`-by-`N`.

For example, to specify

x1 + 2x2 + 3x3 ≤ 10
2x1 + 4x2 + x3 ≤ 20,

give these constraints:

```A = [1,2,3;2,4,1]; b = [10;20];```

Example: To specify that the control variables sum to 1, give the constraints `Aeq = ones(1,N)` and ```beq = 1```

Data Types: `double`

`lb` — Lower boundsreal vector | real array

Lower bounds, specified as a real vector or real array. If the number of elements in `x0` is equal to that of `lb`, then `lb` specifies that

`x(i) >= lb(i)`

for all `i`.

If `numel(lb) < numel(x0)`, then `lb` specifies that

`x(i) >= lb(i)`

for

`1 <= i <= numel(lb)`

In this case, solvers issue a warning.

Example: To specify that all control variables are positive, ```lb = zeros(size(x0))```

Data Types: `double`

`ub` — Upper boundsreal vector | real array

Upper bounds, specified as a real vector or real array. If the number of elements in `x0` is equal to that of `ub`, then `ub` specifies that

`x(i) <= ub(i)`

for all `i`.

If `numel(ub) < numel(x0)`, then `ub` specifies that

`x(i) <= ub(i)`

for

`1 <= i <= numel(ub)`

In this case, solvers issue a warning.

Example: To specify that all control variables are less than one, `ub = ones(size(x0))`

Data Types: `double`

`nonlcon` — Nonlinear constraintsfunction handle | function name

Nonlinear constraints, specified as a function handle or function name. `nonlcon` is a function that accepts a vector or array `x` and returns two arrays, `c(x)` and `ceq(x)`.

• `c(x)` is the array of nonlinear inequality constraints at `x`. `patternsearch` attempts to satisfy

`c(x) <= 0`

for all entries of `c`.

• `ceq(x)` is the array of nonlinear equality constraints at `x`. `patternsearch` attempts to satisfy

`ceq(x) = 0`

for all entries of `ceq`.

For example,

`x = patternsearch(@myfun,x0,A,b,Aeq,beq,lb,ub,@mycon)`

where `mycon` is a MATLAB function such as

```function [c,ceq] = mycon(x) c = ... % Compute nonlinear inequalities at x. ceq = ... % Compute nonlinear equalities at x.```
For more information, see Nonlinear Constraints .

Data Types: `char` | `function_handle`

`options` — Optimization optionsstructure such as the one returned by `psoptimset`

Optimization options, specified as a structure such as the one returned by `psoptimset`. For details, see Pattern Search Options.

OptionDescriptionValues
`Cache`

With `Cache` set to `'on'`, `patternsearch` keeps a history of the mesh points it polls. At subsequent iterations, `patternsearch` does not poll points close to those it already polled. Use this option if `patternsearch` runs slowly while computing the objective function. If the objective function is stochastic, do not to use this option.

`'on'` | `{'off'}`
`CacheSize`

Size of the history.

Positive scalar | `{1e4}`

`CacheTol`

Largest distance from the current mesh point to any point in the history in order for `patternsearch` to avoid polling the current point. Use if `'Cache'` option is set to `'on'`.

Positive scalar | `{eps}`

`CompletePoll`

Complete poll around current point. See How Pattern Search Polling Works.

`'on'` | `{'off'}`
`CompleteSearch`

Complete search around current point when the search method is a poll method. See Searching and Polling.

`'on'` | `{'off'}`
`Display`

Level of display.

`'off'` | `'iter'` | `'diagnose'` | `{'final'}`
`InitialMeshSize`

Initial mesh size for pattern algorithm. See How Pattern Search Polling Works.

Positive scalar | `{1.0}`

`InitialPenalty`

Initial value of the penalty parameter. See Nonlinear Constraint Solver Algorithm.

Positive scalar | `{10}`

`MaxFunEvals`

Maximum number of objective function evaluations.

Positive integer | `{2000*numberOfVariables}`

`MaxIter`

Maximum number of iterations.

Positive integer | `{100*numberOfVariables}`

`MaxMeshSize`

Maximum mesh size used in a poll or search step. See How Pattern Search Polling Works.

Positive scalar | `{Inf}`

`MeshAccelerator`

Accelerate convergence near a minimum. `patternsearch` contracts by an extra factor of 1/2.

`'on'` | {`'off'`}
`MeshContraction`

Mesh contraction factor for unsuccessful iteration.

Positive scalar | `{0.5}`
`MeshExpansion`

Mesh expansion factor for successful iteration.

Positive scalar | `{2.0}`

`MeshRotate`

Rotate the pattern before declaring a point to be optimum. See Mesh Options.

`'off'` | `{'on'}`
`OutputFcns`

Specifies a user-defined function that an optimization function calls at each iteration.

Function handle or cell array of function handles | `{[]}`

`PenaltyFactor`

Penalty update parameter. See Nonlinear Constraint Solver Algorithm.

Positive scalar | `{100}`

`PlotFcns`

Specifies plots of output from the pattern search.

`{[]}` | `@psplotbestf` | `@psplotmeshsize` | `@psplotfuncount` | `@psplotbestx` | custom plot function

`PlotInterval`

Specifies that plot functions will be called at every interval.

`{1}` | positive integer

`PollingOrder`

Order of poll directions in pattern search .

`'Random'` | `'Success'` | `{'Consecutive'}`

`PollMethod`

Polling strategy used in pattern search.

`{'GPSPositiveBasis2N'}` ` |` `'GPSPositiveBasisNp1'`` |` `'GSSPositiveBasis2N'`` |` `'GSSPositiveBasisNp1'`` |` `'MADSPositiveBasis2N'`` |` `'MADSPositiveBasisNp1'`

`ScaleMesh`

Automatic scaling of variables.

{`'on'`} | `'off'`
`SearchMethod`

Type of search used in pattern search.

```{[]} | @GPSPositiveBasis2N | @GPSPositiveBasisNp1 | @GSSPositiveBasis2N | @GSSPositiveBasisNp1 | @MADSPositiveBasis2N | @MADSPositiveBasisNp1 | @searchga | @searchlhs | @searchneldermead | ```custom search function
`TimeLimit`

Total time (in seconds) allowed for optimization.

Positive scalar | `{Inf}`

`TolBind`

Binding tolerance. See Constraint Parameters.

Positive scalar | `{1e-3}`

`TolCon`

Tolerance on constraints.

Positive scalar | `{1e-6}`

`TolFun`

Tolerance on function. Iterations stop if the change in function value is less than `TolFun` and the mesh size is less than `TolX`. Does not apply to MADS polling.

Positive scalar | `{1e-6}`

`TolMesh`

Tolerance on mesh size.

Positive scalar | `{1e-6}`

`TolX`

Tolerance on variable. Iterations stop if both the change in position and the mesh size are less than `TolX`. Does not apply to MADS polling.

Positive scalar | `{1e-6}`

`UseParallel`

Compute objective and nonlinear constraint functions in parallel. See Vectorize and Parallel Options (User Function Evaluation) and How to Use Parallel Processing.

`true` | `{false}`

`Vectorized`

Specifies whether functions are vectorized. See Vectorize and Parallel Options (User Function Evaluation) and Vectorize the Objective and Constraint Functions.

`'on'` | `{'off'}`

Example: `options = psoptimset('MaxIter',150,'TolMesh',1e-4)`

Data Types: `struct`

`problem` — Problem structurestructure

Problem structure, specified as a structure with the following fields:

• `objective` — Objective function

• `x0` — Starting point

• `Aineq` — Matrix for linear inequality constraints

• `bineq` — Vector for linear inequality constraints

• `Aeq` — Matrix for linear equality constraints

• `beq` — Vector for linear equality constraints

• `lb` — Lower bound for `x`

• `ub` — Upper bound for `x`

• `nonlcon` — Nonlinear constraint function

• `solver``'patternsearch'`

• `options` — Options structure created with `psoptimset`

• `rngstate` — Optional field to reset the state of the random number generator

Create the structure `problem` by exporting a problem from the Optimization app, as described in Importing and Exporting Your Work.

 Note   All fields in `problem` are required.

Data Types: `struct`

Output Arguments

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`x` — Solutionreal vector

Solution, returned as a real vector. The size of `x` is the same as the size of `x0`. When `exitflag` is positive, `x` is typically a local solution to the problem.

`fval` — Objective function value at the solutionreal number

Objective function value at the solution, returned as a real number. Generally, `fval` = `fun(x)`.

`exitflag` — Reason `patternsearch` stoppedinteger

Reason `patternsearch` stopped, returned as an integer.

Exit FlagMeaning

`1`

Without nonlinear constraints — The magnitude of the mesh size is less than the specified tolerance, and the constraint violation is less than `TolCon`.

With nonlinear constraints — The magnitude of the complementarity measure (defined after this table) is less than `sqrt(TolCon)`, the subproblem is solved using a mesh finer than `TolMesh`, and the constraint violation is less than `TolCon`.

`2`

The change in `x` and the mesh size are both less than the specified tolerance, and the constraint violation is less than `TolCon`.

`3`

The change in `fval` and the mesh size are both less than the specified tolerance, and the constraint violation is less than `TolCon`.

`4`

The magnitude of the step is smaller than machine precision, and the constraint violation is less than `TolCon`.

`0`

The maximum number of function evaluations or iterations is reached.

`-1`

Optimization terminated by an output function or plot function.

`-2`

No feasible point found.

In the nonlinear constraint solver, the complementarity measure is the norm of the vector whose elements are ciλi, where ci is the nonlinear inequality constraint violation, and λi is the corresponding Lagrange multiplier.

`output` — Information about the optimization processstructure

Information about the optimization process, returned as a structure with these fields:

• `function` — Objective function.

• `problemtype` — String describing the type of problem:

• `'unconstrained'`

• `'boundconstraints'`

• `'linearconstraints'`

• `'nonlinearconstr'`

• `pollmethod` — Polling technique.

• `searchmethod` — Search technique used, if any.

• `iterations` — Total number of iterations.

• `funccount` — Total number of function evaluations.

• `meshsize` — Mesh size at `x`.

• `maxconstraint` — Maximum constraint violation, if any.

• `rngstate` — State of the MATLAB random number generator, just before the algorithm started. You can use the values in `rngstate` to reproduce the output when you use a random search method or random poll method. See Reproduce Results, which discusses the identical technique for `ga`.

• `message` — Reason why the algorithm terminated.

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Algorithms

By default, `patternsearch` looks for a minimum based on an adaptive mesh that, in the absence of linear constraints, is aligned with the coordinate directions. See What Is Direct Search? and How Pattern Search Polling Works.

References

[1] Audet, Charles, and J. E. Dennis Jr. "Analysis of Generalized Pattern Searches." SIAM Journal on Optimization. Volume 13, Number 3, 2003, pp. 889–903.

[2] Conn, A. R., N. I. M. Gould, and Ph. L. Toint. "A Globally Convergent Augmented Lagrangian Barrier Algorithm for Optimization with General Inequality Constraints and Simple Bounds." Mathematics of Computation. Volume 66, Number 217, 1997, pp. 261–288.

[3] Abramson, Mark A. Pattern Search Filter Algorithms for Mixed Variable General Constrained Optimization Problems. Ph.D. Thesis, Department of Computational and Applied Mathematics, Rice University, August 2002.

[4] Abramson, Mark A., Charles Audet, J. E. Dennis, Jr., and Sebastien Le Digabel. "ORTHOMADS: A deterministic MADS instance with orthogonal directions." SIAM Journal on Optimization. Volume 20, Number 2, 2009, pp. 948–966.

[5] Kolda, Tamara G., Robert Michael Lewis, and Virginia Torczon. "Optimization by direct search: new perspectives on some classical and modern methods." SIAM Review. Volume 45, Issue 3, 2003, pp. 385–482.

[6] Kolda, Tamara G., Robert Michael Lewis, and Virginia Torczon. "A generating set direct search augmented Lagrangian algorithm for optimization with a combination of general and linear constraints." Technical Report SAND2006-5315, Sandia National Laboratories, August 2006.

[7] Lewis, Robert Michael, Anne Shepherd, and Virginia Torczon. "Implementing generating set search methods for linearly constrained minimization." SIAM Journal on Scientific Computing. Volume 29, Issue 6, 2007, pp. 2507–2530.