OptimizationFunction

The OptimizationFunction type is a function type that holds all of the extra differentiation data required to do fast and accurate optimization. The signature for the constructor is:

OptimizationFunction{iip}(f,adtype=NoAD();
                          grad=nothing,
                          hess=nothing,
                          hv=nothing,
                          cons=nothing,
                          cons_j=nothing,
                          cons_h=nothing)

The keyword arguments are as follows:

• grad: Gradient
• hess: Hessian
• hv: Hessian vector products hv(du,u,p,t,v) = H*v
• cons: Constraint function
• cons_j
• cons_h

Defining Optimization Functions Via AD

While using the keyword arguments gives the user control over defining all of the possible functions, the simplest way to handle the generation of an OptimizationFunction is by specifying an AD type. By doing so, this will automatically fill in all of the extra functions. For example,

OptimizationFunction(f,AutoZygote())

will use Zygote.jl to define all of the necessary functions. Note that if any functions are defined directly, the auto-AD definition does not overwrite the user's choice.

The choices for the auto-AD fill-ins with quick descriptions are:

• AutoForwardDiff(): The fastest choice for small optimizations
• AutoReverseDiff(compile=false): A fast choice for large scalar optimizations
• AutoTracker(): Like ReverseDiff but GPU-compatible
• AutoZygote(): The fastest choice
• AutoFiniteDiff(): Finite differencing, not optimal but always applicable
• AutoModelingToolkit(): The fastest choice for large scalar optimizations

The following sections describe the Auto-AD choices in detail.

AutoForwardDiff

This uses the ForwardDiff.jl package. It is the fastest choice for small systems, especially with heavy scalar interactions. It is easy to use and compatible with most pure is Julia functions which have loose type restrictions. However, because it's forward-mode, it scales poorly in comparison to other AD choices. Hessian construction is suboptimal as it uses the forward-over-forward approach.

• Compatible with GPUs
• Compatible with Hessian-based optimization
• Compatible with Hv-based optimization
• Compatible with constraints

AutoReverseDiff

This uses the ReverseDiff.jl package. AutoReverseDiff has a default argument, compile, which denotes whether the reverse pass should be compiled. compile should only be set to true if f contains no branches (if statements, while loops) otherwise it can produce incorrect derivatives!.

AutoReverseDiff is generally applicable to many pure Julia codes, and with compile=true it is one of the fastest options on code with heavy scalar interactions. Hessian calculations are fast by mixing ForwardDiff with ReverseDiff for forward-over-reverse. However, its performance can falter when compile=false.

• Not compatible with GPUs
• Compatible with Hessian-based optimization by mixing with ForwardDiff
• Compatible with Hv-based optimization by mixing with ForwardDiff
• Not compatible with constraint functions

AutoTracker

This uses the Tracker.jl package. Generally slower than ReverseDiff, it is generally applicable to many pure Julia codes.

• Compatible with GPUs
• Not compatible with Hessian-based optimization
• Not compatible with Hv-based optimization
• Not compatible with constraint functions

AutoZygote

This uses the Zygote.jl package. This is the staple reverse-mode AD that handles a large portion of Julia with good efficiency. Hessian construction is fast via forward-over-reverse mixing ForwardDiff.jl with Zygote.jl

• Compatible with GPUs
• Compatible with Hessian-based optimization via ForwardDiff
• Compatible with Hv-based optimization via ForwardDiff
• Not compatible with constraint functions

AutoFiniteDiff

This uses FiniteDiff.jl. While to necessarily the most efficient in any case, this is the only choice that doesn't require the f function to be automatically differentiable, which means it applies to any choice. However, because it's using finite differencing, one needs to be careful as this procedure introduces numerical error into the derivative estimates.

• Compatible with GPUs
• Compatible with Hessian-based optimization
• Compatible with Hv-based optimization
• Not compatible with constraint functions

AutoModelingToolkit

This uses the ModelingToolkit.jl symbolic system for automatically converting the f function into a symbolic equation and uses symbolic differentiation in order to generate a fast derivative code. Note that this will also compile a new version of your f function that is automatically optimized. In this choice, it defaults to grad=false and hess=false, and one must change these to true in order to enable the symbolic derivation. Future updates will enable automatic parallelization and sparsity in the derived functions. This can be the fastest for many systems, especially when parallelization and sparsity are required, but can take the longest to generate.

• Not compatible with GPUs
• Compatible with Hessian-based optimization
• Not compatible with Hv-based optimization
• Not compatible with constraint functions