# OptimLib

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

Features:

- A C++11 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:
- OpenMP-accelerated algorithms for parallel computation.
- Straightforward linking with parallelized BLAS libraries, such as OpenBLAS.
- 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 Method 1: Shared Library
- Installation Method 2: Header-only Library
- R Compatibility
- Examples
- 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(Vec_t& init_out_vals, std::function<double (const Vec_t& vals_inp, Vec_t* grad_out, void* opt_data)> opt_objfn, void* opt_data);`

where `Vec_t`

is used to represent either `arma::vec`

or `Eigen::VectorXd`

types.

## Installation Method 1: Shared Library

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

method:

```
# clone optim into the current directory
git clone https://github.com/kthohr/optim ./optim
# build and install
cd ./optim
./configure -i "/usr/local" -l arma -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`-l`

specify the choice of linear algebra library;`arma`

or`eigen`

`-m`

specify the BLAS and Lapack libraries to link against; 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)

### Linear Algebra Library

OptimLib requires either the Armadillo or Eigen C++ linear algebra libraries.

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

:

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

## 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 add the compiler definition `USE_RCPP_ARMADILLO`

before including the OptimLib files:

```
#define USE_RCPP_ARMADILLO
#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_ARMA_WRAPPERS
#include "optim.hpp"
//
// Ackley function
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 pi = arma::datum::pi;
double obj_val = -20*std::exp( -0.2*std::sqrt(0.5*(x*x + y*y)) ) - std::exp( 0.5*(std::cos(2*pi*x) + std::cos(2*pi*y)) ) + 22.718282L;
//
return obj_val;
}
int main()
{
// initial values:
arma::vec x = arma::ones(2,1) + 1.0; // (2,2)
//
std::chrono::time_point<std::chrono::system_clock> start = std::chrono::system_clock::now();
bool success = optim::de(x,ackley_fn,nullptr);
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 << "de: Ackley test completed successfully.\n"
<< "elapsed time: " << elapsed_seconds.count() << "s\n";
} else {
std::cout << "de: Ackley test completed unsuccessfully." << std::endl;
}
arma::cout << "\nde: 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
```

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.

Check the `/tests`

directory for additional examples, and http://www.kthohr.com/optimlib.html 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
```

## Author

Keith O'Hara

## License

Apache Version 2