NLopt.jl

NLopt.jl

NLopt.jl is a wrapper for the NLopt library for nonlinear optimization.

NLopt provides a common interface for many different optimization algorithms, including:

• Both global and local optimization
• Algorithms using function values only (derivative-free) and also algorithms exploiting user-supplied gradients.
• Algorithms for unconstrained optimization, bound-constrained optimization, and general nonlinear inequality/equality constraints.

License

NLopt.jl is licensed under the MIT License.

The underlying solver, stevengj/nlopt, is licensed under the LGPL v3.0 license.

Installation

Install NLopt.jl using the Julia package manager:

import Pkg
Pkg.add("NLopt")

In addition to installing the NLopt.jl package, this will also download and install the NLopt binaries. You do not need to install NLopt separately.

Tutorial

The following example code solves the nonlinearly constrained minimization problem from the NLopt Tutorial.

using NLopt
function my_objective_fn(x::Vector, grad::Vector)
    if length(grad) > 0
        grad[1] = 0
        grad[2] = 0.5 / sqrt(x[2])
    end
    return sqrt(x[2])
end
function my_constraint_fn(x::Vector, grad::Vector, a, b)
    if length(grad) > 0
        grad[1] = 3 * a * (a * x[1] + b)^2
        grad[2] = -1
    end
    return (a * x[1] + b)^3 - x[2]
end
opt = NLopt.Opt(:LD_MMA, 2)
NLopt.lower_bounds!(opt, [-Inf, 0.0])
NLopt.xtol_rel!(opt, 1e-4)
NLopt.min_objective!(opt, my_objective_fn)
NLopt.inequality_constraint!(opt, (x, g) -> my_constraint_fn(x, g, 2, 0), 1e-8)
NLopt.inequality_constraint!(opt, (x, g) -> my_constraint_fn(x, g, -1, 1), 1e-8)
min_f, min_x, ret = NLopt.optimize(opt, [1.234, 5.678])
num_evals = NLopt.numevals(opt)
println(
    """
    objective value       : $min_f
    solution              : $min_x
    solution status       : $ret
    # function evaluation : $num_evals
    """
)

The output is:

objective value       : 0.5443310477213124
solution              : [0.3333333342139688, 0.29629628951338166]
solution status       : XTOL_REACHED
# function evaluation : 11

Algorithm-specific parameters

Some algorithms have algorithm-specific parameters that can be set via NLopt.nlopt_set_param. For example:

julia> import NLopt

julia> opt = NLopt.Opt(:LD_MMA, 2)
Opt(LD_MMA, 2)

julia> NLopt.nlopt_set_param(opt, "inner_maxeval", 5)
NLOPT_SUCCESS::nlopt_result = 1

Consult the NLopt documentation for the list of parameters supported by each algorithm.

Trace iterations

A common feature request is for a callback that can used to trace the solution over the iterations of the optimizer.

There is no native support for this in NLopt. Instead, add the callback to your objective function.

julia> using NLopt

julia> begin
           trace = Any[]
           function my_objective_fn(x::Vector, grad::Vector)
               if length(grad) > 0
                   grad[1] = 0
                   grad[2] = 0.5 / sqrt(x[2])
               end
               value = sqrt(x[2])
               push!(trace, copy(x) => value)
               return value
           end
           function my_constraint_fn(x::Vector, grad::Vector, a, b)
               if length(grad) > 0
                   grad[1] = 3 * a * (a * x[1] + b)^2
                   grad[2] = -1
               end
               return (a * x[1] + b)^3 - x[2]
           end
           opt = NLopt.Opt(:LD_MMA, 2)
           NLopt.lower_bounds!(opt, [-Inf, 0.0])
           NLopt.xtol_rel!(opt, 1e-4)
           NLopt.min_objective!(opt, my_objective_fn)
           NLopt.inequality_constraint!(opt, (x, g) -> my_constraint_fn(x, g, 2, 0), 1e-8)
           NLopt.inequality_constraint!(opt, (x, g) -> my_constraint_fn(x, g, -1, 1), 1e-8)
           min_f, min_x, ret = NLopt.optimize(opt, [1.234, 5.678])
       end
(0.5443310477213124, [0.3333333342139688, 0.29629628951338166], :XTOL_REACHED)

julia> trace
11-element Vector{Any}:
                            [1.234, 5.678] => 2.382855429941145
   [0.8787394664016357, 5.551370325142423] => 2.3561346152421816
   [0.8262160034228196, 5.043903787432386] => 2.245863706334912
  [0.4739440370386794, 4.0767726724255375] => 2.0191019470114773
    [0.35389779634506047, 3.0308503583016] => 1.7409337604577608
 [0.33387310647853335, 1.9717933962872487] => 1.4042056104029954
  [0.3333337209575201, 1.0450874902862517] => 1.0222952070152005
 [0.33333357431034494, 0.4695027039311135] => 0.6852026736164369
  [0.3333332772332185, 0.3057923933552822] => 0.5529849847466767
 [0.33333339455750244, 0.2963215980646768] => 0.5443542946139737
 [0.3333333342139688, 0.29629628951338166] => 0.5443310477213124

Use with JuMP

NLopt implements the MathOptInterface interface for nonlinear optimization, which means that it can be used interchangeably with other optimization packages from modeling packages like JuMP. Note that NLopt does not exploit sparsity of Jacobians.

You can use NLopt with JuMP as follows:

using JuMP, NLopt
model = Model(NLopt.Optimizer)
set_attribute(model, "algorithm", :LD_MMA)
set_attribute(model, "xtol_rel", 1e-4)
set_attribute(model, "constrtol_abs", 1e-8)
@variable(model, x[1:2])
set_lower_bound(x[2], 0.0)
set_start_value.(x, [1.234, 5.678])
@NLobjective(model, Min, sqrt(x[2]))
@NLconstraint(model, (2 * x[1] + 0)^3 - x[2] <= 0)
@NLconstraint(model, (-1 * x[1] + 1)^3 - x[2] <= 0)
optimize!(model)
min_f, min_x, ret = objective_value(model), value.(x), raw_status(model)
println(
    """
    objective value       : $min_f
    solution              : $min_x
    solution status       : $ret
    """
)

The output is:

objective value       : 0.5443310477213124
solution              : [0.3333333342139688, 0.29629628951338166]
solution status       : XTOL_REACHED

The algorithm attribute is required. The value must be one of the supported NLopt algorithms.

Other parameters include stopval, ftol_rel, ftol_abs, xtol_rel, xtol_abs, constrtol_abs, maxeval, maxtime, initial_step, population, seed, and vector_storage.

The algorithm parameter is required, and all others are optional. The meaning and acceptable values of all parameters, except constrtol_abs, match the descriptions below from the specialized NLopt API.

The constrtol_abs parameter is an absolute feasibility tolerance applied to all constraints.

Automatic differentiation

Some algorithms in NLopt require derivatives, which you must manually provide in the if length(grad) > 0 branch of your objective and constraint functions.

To stay simple and lightweight, NLopt does not provide ways to automatically compute derivatives. If you do not have analytic expressions for the derivatives, use a package such as ForwardDiff.jl to compute automatic derivatives.

Here is an example of how to wrap a function f(x::Vector) using ForwardDiff so that it is compatible with NLopt:

using NLopt
import ForwardDiff
function autodiff(f::Function)
    function nlopt_fn(x::Vector, grad::Vector)
        if length(grad) > 0
            # Use ForwardDiff to compute the gradient. Replace with your
            # favorite Julia automatic differentiation package.
            ForwardDiff.gradient!(grad, f, x)
        end
        return f(x)
    end
end
# These functions do not implement `grad`:
my_objective_fn(x::Vector) = sqrt(x[2]);
my_constraint_fn(x::Vector, a, b) = (a * x[1] + b)^3 - x[2];
opt = NLopt.Opt(:LD_MMA, 2)
NLopt.lower_bounds!(opt, [-Inf, 0.0])
NLopt.xtol_rel!(opt, 1e-4)
# But we wrap them in autodiff before passing to NLopt:
NLopt.min_objective!(opt, autodiff(my_objective_fn))
NLopt.inequality_constraint!(opt, autodiff(x -> my_constraint_fn(x, 2, 0)), 1e-8)
NLopt.inequality_constraint!(opt, autodiff(x -> my_constraint_fn(x, -1, 1)), 1e-8)
min_f, min_x, ret = NLopt.optimize(opt, [1.234, 5.678])
# (0.5443310477213124, [0.3333333342139688, 0.29629628951338166], :XTOL_REACHED)

Reference

The main purpose of this section is to document the syntax and unique features of the Julia interface. For more detail on the underlying features, please refer to the C documentation in the NLopt Reference.

Using the Julia API

To use NLopt in Julia, your Julia program should include the line:

using NLopt

which imports the NLopt module and its symbols. Alternatively, you can use import NLopt if you want to keep all the NLopt symbols in their own namespace. You would then prefix all functions below with NLopt., for example NLopt.Opt and so on.

The Opt type

The NLopt API revolves around an object of type Opt.

The object should normally be created via the constructor:

opt = Opt(algorithm::Symbol, n::Int)

given an algorithm (see NLopt Algorithms for possible values) and the dimensionality of the problem (n, the number of optimization parameters).

Whereas in C the algorithms are specified by nlopt_algorithm constants of the form like NLOPT_LD_MMA, the Julia algorithm values are symbols of the form :LD_MMA with the NLOPT_ prefix replaced by : to create a Julia symbol.

There is also a copy(opt::Opt) function to make a copy of a given object (equivalent to nlopt_copy in the C API).

If there is an error in these functions, an exception is thrown.

The algorithm and dimension parameters of the object are immutable (cannot be changed without constructing a new object). Query them using:

ndims(opt::Opt)
algorithm(opt::Opt)

Get a string description of the algorithm via:

algorithm_name(opt::Opt)

Objective function

The objective function is specified by calling one of:

min_objective!(opt::Opt, f::Function)
max_objective!(opt::Opt, f::Function)

depending on whether one wishes to minimize or maximize the objective function f, respectively.

The function f must be of the form:

function f(x::Vector{Float64}, grad::Vector{Float64})
    if length(grad) > 0
        ...set grad to gradient, in-place...
    end
    return ...value of f(x)...
end

The return value must be the value of the function at the point x, where x is a Vector{Float64} array of length n of the optimization parameters.

In addition, if the argument grad is not empty (that is, length(grad) > 0), then grad is a Vector{Float64} array of length n which should (upon return) be set to the gradient of the function with respect to the optimization parameters at x.

Not all of the optimization algorithms (below) use the gradient information: for algorithms listed as “derivative-free,” the grad argument will always be empty and need never be computed. For algorithms that do use gradient information, grad may still be empty for some calls.

Note that grad must be modified in-place by your function f. Generally, this means using indexing operations grad[...] = ... to overwrite the contents of grad. For example grad = 2x will not work, because it points grad to a new array 2x rather than overwriting the old contents; instead, use an explicit loop or use grad[:] = 2x.

Bound constraints

Add bound constraints with:

lower_bounds!(opt::Opt, lb::Union{AbstractVector,Real})
upper_bounds!(opt::Opt, ub::Union{AbstractVector,Real})

where lb and ub are real arrays of length n (the same as the dimension passed to the Opt constructor).

For convenience, you can instead use a single scalar for lb or ub in order to set the lower/upper bounds for all optimization parameters to a single constant.

To retrieve the values of the lower or upper bounds, use:

lower_bounds(opt::Opt)
upper_bounds(opt::Opt)

both of which return Vector{Float64} arrays.

To specify an unbounded dimension, you can use Inf or -Inf.

Nonlinear constraints

Specify nonlinear inequality and equality constraints by the functions:

inequality_constraint!(opt::Opt, f::Function, tol::Real = 0.0)
equality_constraint!(opt::Opt, f::Function, tol::Real = 0.0)

where the arguments f have the same form as the objective function above.

The optional tol arguments specify a tolerance (which defaults to zero) that is used to judge feasibility for the purposes of stopping the optimization.

Each call to these function adds a new constraint to the set of constraints, rather than replacing the constraints.

Remove all of the inequality and equality constraints from a given problem with:

remove_constraints!(opt::Opt)

Vector-valued constraints

Specify vector-valued nonlinear inequality and equality constraints by the functions:

inequality_constraint!(opt::Opt, f::Function, tol::AbstractVector)
equality_constraint!(opt::Opt, f::Function, tol::AbstractVector)

where tol is an array of the tolerances in each constraint dimension; the dimensionality m of the constraint is determined by length(tol).

The constraint function f must be of the form:

function f(result::Vector{Float64}, x::Vector{Float64}, grad::Matrix{Float64})
    if length(grad) > 0
        ...set grad to gradient, in-place...
    end
    result[1] = ...value of c1(x)...
    result[2] = ...value of c2(x)...
    return

where result is a Vector{Float64} array whose length equals the dimensionality m of the constraint (same as the length of tol above), which upon return, should be set in-place to the constraint results at the point x. Any return value of the function is ignored.

In addition, if the argument grad is not empty (that is, length(grad) > 0), then grad is a matrix of size n×m which should (upon return) be set in-place (see above) to the gradient of the function with respect to the optimization parameters at x. That is, grad[j,i] should upon return contain the partial derivative ∂f_i/∂x_j.

A full example is:

julia> using NLopt

julia> function my_objective_fn(x::Vector, grad::Vector)
           if length(grad) > 0
               grad[1] = 0
               grad[2] = 0.5 / sqrt(x[2])
           end
           return sqrt(x[2])
       end
my_objective_fn (generic function with 1 method)

julia> function my_constraint_fn(x::Vector, grad::Vector, a, b)
           if length(grad) > 0
               grad[1] = 3 * a * (a * x[1] + b)^2
               grad[2] = -1
           end
           return (a * x[1] + b)^3 - x[2]
       end
my_constraint_fn (generic function with 1 method)

julia> function constraints(result::Vector, x::Vector, grad::Matrix)
           ∇g1 = [0.0, 0.0]
           ∇g2 = [0.0, 0.0]
           result[1] = my_constraint_fn(x, ∇g1, 2, 0)
           result[2] = my_constraint_fn(x, ∇g2, -1, 1)
           if length(grad) > 0
               # grad has shape (num variables, num constraints)
               grad[:, 1] .= ∇g1
               grad[:, 2] .= ∇g2
           end
           return
       end
constraints (generic function with 1 method)

julia> opt = NLopt.Opt(:LD_MMA, 2)
Opt(LD_MMA, 2)

julia> NLopt.lower_bounds!(opt, [-Inf, 0.0])

julia> NLopt.xtol_rel!(opt, 1e-6)

julia> NLopt.min_objective!(opt, my_objective_fn)

julia> NLopt.inequality_constraint!(opt, constraints, fill(1e-8, 2))

julia> min_f, min_x, ret = NLopt.optimize(opt, [1.234, 5.678])
(0.5443310477213124, [0.3333333342139688, 0.29629628951338166], :XTOL_REACHED)

julia> num_evals = NLopt.numevals(opt)
18

julia> println(
           """
           objective value       : $min_f
           solution              : $min_x
           solution status       : $ret
           # function evaluation : $num_evals
           """
       )
objective value       : 0.5443310477213124
solution              : [0.3333333342139688, 0.29629628951338166]
solution status       : XTOL_REACHED
# function evaluation : 18

Not all of the optimization algorithms (below) use the gradient information: for algorithms listed as “derivative-free,” the grad argument will always be empty and need never be computed. For algorithms that do use gradient information, grad may still be empty for some calls.

You can add multiple vector-valued constraints and/or scalar constraints in the same problem.

Stopping criteria

As explained in the C API Reference and the Introduction, you have multiple options for different stopping criteria that you can specify. (Unspecified stopping criteria are disabled; that is, they have innocuous defaults.)

For each stopping criteria, there are two functions that you can use to get and set the value of the stopping criterion.

stopval(opt::Opt)          # return the current value of `stopval`
stopval!(opt::Opt, value)  # set stopval to `value`

Stop when an objective value of at least stopval is found. (Defaults to -Inf.)

ftol_rel(opt::Opt)
ftol_rel!(opt::Opt, value)

Relative tolerance on function value. (Defaults to 0.)

ftol_abs(opt::Opt)
ftol_abs!(opt::Opt, value)

Absolute tolerance on function value. (Defaults to 0.)

xtol_rel(opt::Opt)
xtol_rel!(opt::Opt, value)

Relative tolerances on the optimization parameters. (Defaults to 0.)

xtol_abs(opt::Opt)
xtol_abs!(opt::Opt, value)

Absolute tolerances on the optimization parameters. (Defaults to 0.)

In the case of xtol_abs, you can either set it to a scalar (to use the same tolerance for all inputs) or a vector of length n (the dimension specified in the Opt constructor) to use a different tolerance for each parameter.

maxeval(opt::Opt)
maxeval!(opt::Opt, value)

Stop when the number of function evaluations exceeds mev. (0 or negative for no limit, which is the default.)

maxtime(opt::Opt)
maxtime!(opt::Opt, value)

Stop when the optimization time (in seconds) exceeds t. (0 or negative for no limit, which is the default.)

Forced termination

In certain cases, the caller may wish to force the optimization to halt, for some reason unknown to NLopt. For example, if the user presses Ctrl-C, or there is an error of some sort in the objective function. You can do this by throwing any exception inside your objective/constraint functions: the optimization will be halted gracefully, and the same exception will be thrown to the caller. The Julia equivalent of nlopt_forced_stop from the C API is to throw a ForcedStop exception.

Performing the optimization

Once all of the desired optimization parameters have been specified in a given object opt::Opt, you can perform the optimization by calling:

optf, optx, ret = optimize(opt::Opt, x::AbstractVector)

On input, x is an array of length n (the dimension of the problem from the Opt constructor) giving an initial guess for the optimization parameters. The return value optx is a array containing the optimized values of the optimization parameters. optf contains the optimized value of the objective function, and ret contains a symbol indicating the NLopt return code (below).

Alternatively:

optf, optx, ret = optimize!(opt::Opt, x::Vector{Float64})

is the same but modifies x in-place (as well as returning optx = x).

Return values

The possible return values are the same as the return values in the C API, except that the NLOPT_ prefix is replaced with :. That is, the return values are like :SUCCESS instead NLOPT_SUCCESS.

Local/subsidiary optimization algorithm

Some of the algorithms, especially MLSL and AUGLAG, use a different optimization algorithm as a subroutine, typically for local optimization. You can change the local search algorithm and its tolerances by setting:

local_optimizer!(opt::Opt, local_opt::Opt)

Here, local_opt is another Opt object whose parameters are used to determine the local search algorithm, its stopping criteria, and other algorithm parameters. (However, the objective function, bounds, and nonlinear-constraint parameters of local_opt are ignored.) The dimension n of local_opt must match that of opt.

This makes a copy of the local_opt object, so you can freely change your original local_opt afterwards without affecting opt.

Initial step size

Just as in the C API, you can set the initial step sizes for derivative-free optimization algorithms with:

initial_step!(opt::Opt, dx::Vector)

Here, dx is an array of the (nonzero) initial steps for each dimension, or a single number if you wish to use the same initial steps for all dimensions.

initial_step(opt::Opt, x::AbstractVector) returns the initial step that will be used for a starting guess of x in optimize(opt, x).

Stochastic population

Just as in the C API, you can get and set the initial population for stochastic optimization with:

population(opt::Opt)
population!(opt::Opt, value)

A population of zero, the default, implies that the heuristic default will be used as decided upon by individual algorithms.

Pseudorandom numbers

For stochastic optimization algorithms, NLopt uses pseudorandom numbers generated by the Mersenne Twister algorithm, based on code from Makoto Matsumoto.

By default, the seed for the random numbers is generated from the system time, so that you will get a different sequence of pseudorandom numbers each time you run your program. If you want to use a “deterministic” sequence of pseudorandom numbers, that is, the same sequence from run to run, you can set the seed by calling:

NLopt.srand(seed::Integer)

To reset the seed based on the system time, you can call NLopt.srand_time().

Normally, you don’t need to call this as it is called automatically. However, it might be useful if you want to “re-randomize” the pseudorandom numbers after calling nlopt.srand to set a deterministic seed.

Vector storage for limited-memory quasi-Newton algorithms

Just as in the C API, you can get and set the number M of stored vectors for limited-memory quasi-Newton algorithms, via integer-valued property

vector_storage(opt::Opt)
vector_storage!(opt::Opt, value)

The default is 0, in which case NLopt uses a heuristic nonzero value as determined by individual algorithms.

Version number

The version number of NLopt is given by the global variable:

NLOPT_VERSION::VersionNumber

where VersionNumber is a built-in Julia type from the Julia standard library.

Thread safety

The underlying NLopt library is threadsafe; however, re-using the same Opt object across multiple threads is not.

As an example, instead of:

using NLopt
opt = Opt(:LD_MMA, 2)
# Define problem
solutions = Vector{Any}(undef, 10)
Threads.@threads for i in 1:10
    # Not thread-safe because `opt` is re-used
    solutions[i] = optimize(opt, rand(2))
end

Do instead:

solutions = Vector{Any}(undef, 10)
Threads.@threads for i in 1:10
    # Thread-safe because a new `opt` is created for each thread
    opt = Opt(:LD_MMA, 2)
    # Define problem
    solutions[i] = optimize(opt, rand(2))
end

Author

This module was initially written by Steven G. Johnson, with subsequent contributions by several other authors (see the git history).