## Optimization

To show how the Evolutionary package can be used, we minimize the Rosenbrock function, a classical test problem for numerical optimization. We'll assume that you've already installed the Evolutionary package using Julia's package manager.

### Objective Function Definition

First, we load Evolutionary and define the Rosenbrock function:

using Evolutionary
f(x) = (1.0 - x)^2 + 100.0 * (x - x^2)^2 # Rosenbrock

There are various ways to define your objective function:

• For single-objective optimization, the objective function has to have one parameter and return a scalar value, e.g.
soofun(x) = x + x
• For multi-objective optimization, the objective function has to return an vector of values, e.g.
moofun(x) = [ x, x+1 ]
• If multi-objective function has one parameter, the resulting value array will be copied. To reduce, additional data copy, the function can be defined with two parameters to perform in-place change of the result, e.g.
function moofun(F, x)
F = x
F = x +1
F
end

### Perform Optimization

Once we've defined this function, we can find the minimizer (the input that minimizes the objective) and the minimum (the value of the objective at the minimizer) using any of our favorite optimization algorithms. With a function defined, we just specify a form of an individual x of the population for an evolutionary algorithm, and call optimize with a starting individual x0 and a particular optimization algorithm, e.g. CMAES():

x0 = [0.0, 0.0];
Evolutionary.optimize(f, x0, CMAES())
Evolutionary.optimizeFunction
optimize(f[, F], indiv, algo[, opts])
optimize(f[, F], constr, algo[, opts])
optimize(f[, F], constr, indiv, algo[, opts])
optimize(f[, F], constr, algo, poplt[, opts])

Perform optimization of the function f using the algorithm algo with the population, composed of the initial population poplt, or individuals similar to the original individual indiv, or generated from the constraints constr, with the options opts.

• For multi-objective optimization, the objective value F must be provided.
source

## Configurable options

There are several options that simply take on some default values if the user doesn't provide any.

### Algorithm options

There are different algorithms available in Evolutionary, and they are all listed below. Notice that the constructors are written without input here, but they generally take keywords to tweak the way they work. See the pages describing each solver for more detail.

### General options

In addition to the algorithm, you can alter the behavior of the optimization procedure by using the following Options keyword arguments:

Evolutionary.OptionsType

There are following options available:

• abstol::Float64: the absolute tolerance used in the convergence test
• reltol::Float64: the relative tolerance used in the convergence test
• successive_f_tol::Integer: the additional number of the iterations of the optimization algorithm after the convergence test is satisfied (default: 10)
• iterations::Integer: the total number of the iterations of the optimization algorithm (default: 1000)
• show_trace::Bool: enable the trace information display during the optimization (default: false).
• store_trace::Bool: enable the trace information capturing during the optimization (default: false). The trace can be accessed by using trace function after optimization is finished.
• show_every::Integer: show every ns successive trace message (default: 1)
• time_limit::Float64: the time limit for the optimization run in seconds. If the value set to NaN then the limit is not set. (default: NaN)
• callback: the callback function that is called after each iteration of the optimization algorithm. The function accepts as parameter a trace dictionary, and must return a Bool value which if true terminates the optimization. (default: nothing)
• parallelization::Symbol: allows parallelization of the population fitness evaluation if set to :thread using multiple threads (default: :serial)
• rng::AbstractRNG: a random number generator object that is used to control generation of random data during the evolutionary optimization (default: Random.default_rng())
source

We currently recommend the statically dispatched interface by using the Evolutionary.Options constructor:

res = Evolutionary.optimize(x->-sum(x),
BitVector(zeros(30)),
GA(selection=uniformranking(5),
mutation=flip, crossover=SPX),
Evolutionary.Options(iterations=10))

## Obtaining results

After we have our results in res object, we can use the API for getting optimization results. This consists of a collection of functions. They are not exported, so they have to be prefixed by Evolutionary.. Say we do the following optimization:

julia> res = Evolutionary.optimize(x->-sum(x), BitVector(zeros(3)), GA())

* Status: success

* Candidate solution
Minimizer:  [false, false, false]
Minimum:    0
Iterations: 11

* Found with
Algorithm: GA[P=50,x=0.8,μ=0.1,ɛ=0]

* Convergence measures
|f(x) - f(x')| = 0.0 ≤ 1.0e-12

* Work counters
Seconds run:   0.0466 (vs limit Inf)
Iterations:    11
f(x) calls:    600

You can inspect the result by using a collection of the auxiliary functions, e.g. the minimizer and minimum of the objective functions, which can be found using

julia> Evolutionary.minimizer(res)
3-element BitVector:
0
0
0

julia> Evolutionary.minimum(res)
0

#### Complete list of functions

An OptimizationResults interface for representing an optimization result.

An implementation of the result object for evolutionary optimizations.

## Trace

When store_trace and/or show_trace options are set to true in the Options object, an optimization trace is either captured and/or shown on the screen. By default, only the current state minimum value is displayed in the trace. In order to extend trace record, you need to override trace! function providing specialize function behavior on one of specific parameters.

Evolutionary.trace!Method
trace!(record::Dict{String,Any}, objfun, state, population, method, options)

Update the trace record. This function allows to supplement an additional information into the optimization algorithm trace by modifying a trace record. It can be overridden by specifying particular parameter types.

source

Commonly, you would define a specializations of a state, population, or method parameters of trace! function, e.g.

function trace!(record::Dict{String,Any}, objfun, state, population, method::CMAES, options)
record["σ"] = state.σ
end

## Parallelization

If the objective function is heavily CPU-bound, it's possible to utilize multiple processors/threads to speed up computations. To enable multi-threading evaluation of the objective function, set parallelization option to :thread in the Options object.