Optimization Toolbox™

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 with optimset; just don't set them both to 'on'.

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

• The option UseParallel is set to 'always'. The default value of this option is 'never'.

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

Currently, UseParallel cannot be accessed using the Optimization Tool GUI.

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Parallel Estimation of Gradients

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

∇f(x) ≈ [(f(x + Δ₁e₁) – f(x))/Δ₁, (f(x + Δ₂e₂) – f(x))/Δ₂,...,(f(x + Δ_ne_n) – f(x))/Δ_n],

where

• f represents objective or constraint functions

• e_i are the unit direction vectors

• Δ_i is the size of a step in the e_i direction

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

fmincon uses the parallel subroutine only with the active-set algorithm. fgoalattain and fminimax always use the parallel subroutine.

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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. Figure 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

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Parallel Computing for Optimization

Using Parallel Computing with fmincon, fgoalattain, and fminimax