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OptimLib  

OptimLib is a lightweight C++ library of numerical optimization methods for nonlinear functions.

Features:

• A C++11/14/17 library of local and global optimization algorithms, as well as root finding techniques.
• Derivative-free optimization using advanced, parallelized metaheuristic methods.
• Constrained optimization routines to handle simple box constraints, as well as systems of nonlinear constraints.
• For fast and efficient matrix-based computation, OptimLib supports the following templated linear algebra libraries:
  • Armadillo
  • Eigen (version >= 3.4.0)
• Automatic differentiation functionality is available through use of the Autodiff library
• OpenMP-accelerated algorithms for parallel computation.
• Straightforward linking with parallelized BLAS libraries, such as OpenBLAS.
• Available as a single precision (float) or double precision (double) library.
• Available as a header-only library, or as a compiled shared library.
• Released under a permissive, non-GPL license.

Contents:

• Algorithms
• Documentation
• General API
• Installation
• R Compatibility
• Examples
• Automatic Differentiation
• Author and License

Algorithms

A list of currently available algorithms includes:

• Broyden's Method (for root finding)
• Newton's method, BFGS, and L-BFGS
• Gradient descent: basic, momentum, Adam, AdaMax, Nadam, NadaMax, and more
• Nonlinear Conjugate Gradient
• Nelder-Mead
• Differential Evolution (DE)
• Particle Swarm Optimization (PSO)

Documentation

Full documentation is available online:

A PDF version of the documentation is available here.

API

The OptimLib API follows a relatively simple convention, with most algorithms called in the following manner:

algorithm_id(<initial/final values>, <objective function>, <objective function data>);

The inputs, in order, are:

• A writable vector of initial values to define the starting point of the algorithm. In the event of successful completion, the initial values will be overwritten by the solution vector.
• The 'objective function' is the user-defined function to be minimized (or zeroed-out in the case of root finding methods).
• The final input is optional: it is any object that contains additional parameters necessary to evaluate the objective function.

For example, the BFGS algorithm is called using

bfgs(ColVec_t& init_out_vals, std::function<double (const ColVec_t& vals_inp, ColVec_t* grad_out, void* opt_data)> opt_objfn, void* opt_data);

where ColVec_t is used to represent, e.g., arma::vec or Eigen::VectorXd types.

Installation

OptimLib is available as a compiled shared library, or as header-only library, for Unix-alike systems only (e.g., popular Linux-based distros, as well as macOS). Use of this library with Windows-based systems, with or without MSVC, is not supported.

Requirements

OptimLib requires either the Armadillo or Eigen C++ linear algebra libraries. (Note that Eigen version 3.4.0 requires a C++14-compatible compiler.)

Before including the header files, define one of the following:

#define OPTIM_ENABLE_ARMA_WRAPPERS
#define OPTIM_ENABLE_EIGEN_WRAPPERS

Example:

#define OPTIM_ENABLE_EIGEN_WRAPPERS
#include "optim.hpp"

Installation Method 1: Shared Library

The library can be installed on Unix-alike systems via the standard ./configure && make method.

First clone the library and any necessary submodules:

# clone optim into the current directory
git clone https://github.com/kthohr/optim ./optim

# change directory
cd ./optim

# clone necessary submodules
git submodule update --init

Set (one) of the following environment variables before running configure:

export ARMA_INCLUDE_PATH=/path/to/armadillo
export EIGEN_INCLUDE_PATH=/path/to/eigen

Finally:

# build and install with Eigen
./configure -i "/usr/local" -l eigen -p
make
make install

The final command will install OptimLib into /usr/local.

Configuration options (see ./configure -h):

      Primary

• -h print help
• -i installation path; default: the build directory
• -f floating-point precision mode; default: double
• -l specify the choice of linear algebra library; choose arma or eigen
• -m specify the BLAS and Lapack libraries to link with; for example, -m "-lopenblas" or -m "-framework Accelerate"
• -o compiler optimization options; defaults to -O3 -march=native -ffp-contract=fast -flto -DARMA_NO_DEBUG
• -p enable OpenMP parallelization features (recommended)

      Secondary

• -c a coverage build (used with Codecov)
• -d a 'development' build
• -g a debugging build (optimization flags set to -O0 -g)

      Special

• --header-only-version generate a header-only version of OptimLib (see below)

Installation Method 2: Header-only Library

OptimLib is also available as a header-only library (i.e., without the need to compile a shared library). Simply run configure with the --header-only-version option:

./configure --header-only-version

This will create a new directory, header_only_version, containing a copy of OptimLib, modified to work on an inline basis. With this header-only version, simply include the header files (#include "optim.hpp) and set the include path to the head_only_version directory (e.g.,-I/path/to/optimlib/header_only_version).

R Compatibility

To use OptimLib with an R package, first generate a header-only version of the library (see above). Then simply add a compiler definition before including the OptimLib files.

• For RcppArmadillo:

#define OPTIM_USE_RCPP_ARMADILLO
#include "optim.hpp"

• For RcppEigen:

#define OPTIM_USE_RCPP_EIGEN
#include "optim.hpp"

Examples

To illustrate OptimLib at work, consider searching for the global minimum of the Ackley function:

This is a well-known test function with many local minima. Newton-type methods (such as BFGS) are sensitive to the choice of initial values, and will perform rather poorly here. As such, we will employ a global search method--in this case: Differential Evolution.

Code:

#define OPTIM_ENABLE_EIGEN_WRAPPERS
#include "optim.hpp"
        
#define OPTIM_PI 3.14159265358979

double 
ackley_fn(const Eigen::VectorXd& vals_inp, Eigen::VectorXd* grad_out, void* opt_data)
{
    const double x = vals_inp(0);
    const double y = vals_inp(1);

    const double obj_val = 20 + std::exp(1) - 20*std::exp( -0.2*std::sqrt(0.5*(x*x + y*y)) ) - std::exp( 0.5*(std::cos(2 * OPTIM_PI * x) + std::cos(2 * OPTIM_PI * y)) );
            
    return obj_val;
}
        
int main()
{
    Eigen::VectorXd x = 2.0 * Eigen::VectorXd::Ones(2); // initial values: (2,2)
        
    bool success = optim::de(x, ackley_fn, nullptr);
        
    if (success) {
        std::cout << "de: Ackley test completed successfully." << std::endl;
    } else {
        std::cout << "de: Ackley test completed unsuccessfully." << std::endl;
    }
        
    std::cout << "de: solution to Ackley test:\n" << x << std::endl;
        
    return 0;
}

On x86-based computers, this example can be compiled using:

g++ -Wall -std=c++14 -O3 -march=native -ffp-contract=fast -I/path/to/eigen -I/path/to/optim/include optim_de_ex.cpp -o optim_de_ex.out -L/path/to/optim/lib -loptim

Output:

de: Ackley test completed successfully.
elapsed time: 0.028167s

de: solution to Ackley test:
  -1.2702e-17
  -3.8432e-16

On a standard laptop, OptimLib will compute a solution to within machine precision in a fraction of a second.

The Armadillo-based version of this example:

#define OPTIM_ENABLE_ARMA_WRAPPERS
#include "optim.hpp"
        
#define OPTIM_PI 3.14159265358979

double 
ackley_fn(const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)
{
    const double x = vals_inp(0);
    const double y = vals_inp(1);

    const double obj_val = 20 + std::exp(1) - 20*std::exp( -0.2*std::sqrt(0.5*(x*x + y*y)) ) - std::exp( 0.5*(std::cos(2 * OPTIM_PI * x) + std::cos(2 * OPTIM_PI * y)) );
            
    return obj_val;
}
        
int main()
{
    arma::vec x = arma::ones(2,1) + 1.0; // initial values: (2,2)
        
    bool success = optim::de(x, ackley_fn, nullptr);
        
    if (success) {
        std::cout << "de: Ackley test completed successfully." << std::endl;
    } else {
        std::cout << "de: Ackley test completed unsuccessfully." << std::endl;
    }
        
    arma::cout << "de: solution to Ackley test:\n" << x << arma::endl;
        
    return 0;
}

Compile and run:

g++ -Wall -std=c++11 -O3 -march=native -ffp-contract=fast -I/path/to/armadillo -I/path/to/optim/include optim_de_ex.cpp -o optim_de_ex.out -L/path/to/optim/lib -loptim
./optim_de_ex.out

Check the /tests directory for additional examples, and https://optimlib.readthedocs.io/en/latest/ for a detailed description of each algorithm.

Logistic regression

For a data-based example, consider maximum likelihood estimation of a logit model, common in statistics and machine learning. In this case we have closed-form expressions for the gradient and hessian. We will employ a popular gradient descent method, Adam (Adaptive Moment Estimation), and compare to a pure Newton-based algorithm.

#define OPTIM_ENABLE_ARMA_WRAPPERS
#include "optim.hpp"

// sigmoid function

inline
arma::mat sigm(const arma::mat& X)
{
    return 1.0 / (1.0 + arma::exp(-X));
}

// log-likelihood function data

struct ll_data_t
{
    arma::vec Y;
    arma::mat X;
};

// log-likelihood function with hessian

double ll_fn_whess(const arma::vec& vals_inp, arma::vec* grad_out, arma::mat* hess_out, void* opt_data)
{
    ll_data_t* objfn_data = reinterpret_cast<ll_data_t*>(opt_data);

    arma::vec Y = objfn_data->Y;
    arma::mat X = objfn_data->X;

    arma::vec mu = sigm(X*vals_inp);

    const double norm_term = static_cast<double>(Y.n_elem);

    const double obj_val = - arma::accu( Y%arma::log(mu) + (1.0-Y)%arma::log(1.0-mu) ) / norm_term;

    //

    if (grad_out)
    {
        *grad_out = X.t() * (mu - Y) / norm_term;
    }

    //

    if (hess_out)
    {
        arma::mat S = arma::diagmat( mu%(1.0-mu) );
        *hess_out = X.t() * S * X / norm_term;
    }

    //

    return obj_val;
}

// log-likelihood function for Adam

double ll_fn(const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)
{
    return ll_fn_whess(vals_inp,grad_out,nullptr,opt_data);
}

//

int main()
{
    int n_dim = 5;     // dimension of parameter vector
    int n_samp = 4000; // sample length

    arma::mat X = arma::randn(n_samp,n_dim);
    arma::vec theta_0 = 1.0 + 3.0*arma::randu(n_dim,1);

    arma::vec mu = sigm(X*theta_0);

    arma::vec Y(n_samp);

    for (int i=0; i < n_samp; i++)
    {
        Y(i) = ( arma::as_scalar(arma::randu(1)) < mu(i) ) ? 1.0 : 0.0;
    }

    // fn data and initial values

    ll_data_t opt_data;
    opt_data.Y = std::move(Y);
    opt_data.X = std::move(X);

    arma::vec x = arma::ones(n_dim,1) + 1.0; // initial values

    // run Adam-based optim

    optim::algo_settings_t settings;

    settings.gd_method = 6;
    settings.gd_settings.step_size = 0.1;

    std::chrono::time_point<std::chrono::system_clock> start = std::chrono::system_clock::now();

    bool success = optim::gd(x,ll_fn,&opt_data,settings);

    std::chrono::time_point<std::chrono::system_clock> end = std::chrono::system_clock::now();
    std::chrono::duration<double> elapsed_seconds = end-start;

    //

    if (success) {
        std::cout << "Adam: logit_reg test completed successfully.\n"
                  << "elapsed time: " << elapsed_seconds.count() << "s\n";
    } else {
        std::cout << "Adam: logit_reg test completed unsuccessfully." << std::endl;
    }

    arma::cout << "\nAdam: true values vs estimates:\n" << arma::join_rows(theta_0,x) << arma::endl;

    //
    // run Newton-based optim

    x = arma::ones(n_dim,1) + 1.0; // initial values

    start = std::chrono::system_clock::now();

    success = optim::newton(x,ll_fn_whess,&opt_data);

    end = std::chrono::system_clock::now();
    elapsed_seconds = end-start;

    //

    if (success) {
        std::cout << "newton: logit_reg test completed successfully.\n"
                  << "elapsed time: " << elapsed_seconds.count() << "s\n";
    } else {
        std::cout << "newton: logit_reg test completed unsuccessfully." << std::endl;
    }

    arma::cout << "\nnewton: true values vs estimates:\n" << arma::join_rows(theta_0,x) << arma::endl;

    return 0;
}

Output:

Adam: logit_reg test completed successfully.
elapsed time: 0.025128s

Adam: true values vs estimates:
   2.7850   2.6993
   3.6561   3.6798
   2.3379   2.3860
   2.3167   2.4313
   2.2465   2.3064

newton: logit_reg test completed successfully.
elapsed time: 0.255909s

newton: true values vs estimates:
   2.7850   2.6993
   3.6561   3.6798
   2.3379   2.3860
   2.3167   2.4313
   2.2465   2.3064

Automatic Differentiation

By combining Eigen with the Autodiff library, OptimLib provides experimental support for automatic differentiation.

Example using forward-mode automatic differentiation with BFGS for the Sphere function:

#define OPTIM_ENABLE_EIGEN_WRAPPERS
#include "optim.hpp"

#include <autodiff/forward/real.hpp>
#include <autodiff/forward/real/eigen.hpp>

//

autodiff::real
opt_fnd(const autodiff::ArrayXreal& x)
{
    return x.cwiseProduct(x).sum();
}

double
opt_fn(const Eigen::VectorXd& x, Eigen::VectorXd* grad_out, void* opt_data)
{
    autodiff::real u;
    autodiff::ArrayXreal xd = x.eval();

    if (grad_out) {
        Eigen::VectorXd grad_tmp = autodiff::gradient(opt_fnd, autodiff::wrt(xd), autodiff::at(xd), u);

        *grad_out = grad_tmp;
    } else {
        u = opt_fnd(xd);
    }

    return u.val();
}

int main()
{
    Eigen::VectorXd x(5);
    x << 1, 2, 3, 4, 5;

    bool success = optim::bfgs(x, opt_fn, nullptr);

    if (success) {
        std::cout << "bfgs: forward-mode autodiff test completed successfully.\n" << std::endl;
    } else {
        std::cout << "bfgs: forward-mode autodiff test completed unsuccessfully.\n" << std::endl;
    }

    std::cout << "solution: x = \n" << x << std::endl;

    return 0;
}

Compile with:

g++ -Wall -std=c++17 -O3 -march=native -ffp-contract=fast -I/path/to/eigen -I/path/to/autodiff -I/path/to/optim/include optim_autodiff_ex.cpp -o optim_autodiff_ex.out -L/path/to/optim/lib -loptim

See the documentation for more details on this topic.

Author

Keith O'Hara

License

Apache Version 2