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# Documentation

## Parallel Computing in Optimization Toolbox Functions

### Parallel Optimization Functionality

Parallel computing is the technique of using multiple processors on a single problem. The reason to use parallel computing is to speed computations.

The Optimization Toolbox™ solvers fmincon, fgoalattain, and fminimax can automatically distribute the numerical estimation of gradients of objective functions and nonlinear constraint functions to multiple processors. These solvers use parallel gradient estimation under the following conditions:

• You have a license for Parallel Computing Toolbox™ software.

• The option GradObj is set to 'off', or, if there is a nonlinear constraint function, the option GradConstr is set to 'off'. Since 'off' is the default value of these options, you don't have to set them; just don't set them both to 'on'.

• Parallel computing is enabled with parpool, a Parallel Computing Toolbox function.

• The option UseParallel is set to true. The default value of this option is false.

When these conditions hold, the solvers compute estimated gradients in parallel.

 Note:   Even when running in parallel, a solver occasionally calls the objective and nonlinear constraint functions serially on the host machine. Therefore, ensure that your functions have no assumptions about whether they are evaluated in serial or parallel.

One subroutine was made parallel in the functions fmincon, fgoalattain, and fminimax: the subroutine that estimates the gradient of the objective function and constraint functions. This calculation involves computing function values at points near the current location x. Essentially, the calculation is

$\nabla f\left(x\right)\approx \left[\frac{f\left(x+{\Delta }_{1}{e}_{1}\right)-f\left(x\right)}{{\Delta }_{1}},\frac{f\left(x+{\Delta }_{2}{e}_{2}\right)-f\left(x\right)}{{\Delta }_{2}},\dots ,\frac{f\left(x+{\Delta }_{n}{e}_{n}\right)-f\left(x\right)}{{\Delta }_{n}}\right],$

where

• f represents objective or constraint functions

• ei are the unit direction vectors

• Δi is the size of a step in the ei direction

To estimate ∇f(x) in parallel, Optimization Toolbox solvers distribute the evaluation of (f(x + Δiei) – f(x))/Δi to extra processors.

#### Parallel Central Differences

You can choose to have gradients estimated by central finite differences instead of the default forward finite differences. The basic central finite difference formula is

$\nabla f\left(x\right)\approx \left[\frac{f\left(x+{\Delta }_{1}{e}_{1}\right)-f\left(x-{\Delta }_{1}{e}_{1}\right)}{2{\Delta }_{1}},\dots ,\frac{f\left(x+{\Delta }_{n}{e}_{n}\right)-f\left(x-{\Delta }_{n}{e}_{n}\right)}{2{\Delta }_{n}}\right].$

This takes twice as many function evaluations as forward finite differences, but is usually much more accurate. Central finite differences work in parallel exactly the same as forward finite differences.

Enable central finite differences by using optimoptions to set the FinDiffType option to 'central'. To use forward finite differences, set the FinDiffType option to 'forward'.

### Nested Parallel Functions

Solvers employ the Parallel Computing Toolbox function parfor to perform parallel estimation of gradients. parfor does not work in parallel when called from within another parfor loop. Therefore, you cannot simultaneously use parallel gradient estimation and parallel functionality within your objective or constraint functions.

Suppose, for example, your objective function userfcn calls parfor, and you wish to call fmincon in a loop. Suppose also that the conditions for parallel gradient evaluation of fmincon, as given in Parallel Optimization Functionality, are satisfied. When parfor Runs In Parallel shows three cases:

1. The outermost loop is parfor. Only that loop runs in parallel.

2. The outermost parfor loop is in fmincon. Only fmincon runs in parallel.

3. The outermost parfor loop is in userfcn. userfcn can use parfor in parallel.

When parfor Runs In Parallel