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<!DOCTYPE html>
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<meta name="author" content="Keith O'Hara">
<meta name="keywords" content="Optimization, C++, C++11, Differential Evolution, Particle Swarm Optimization, Root Finding, OpenMP, Parallel Optimization, BFGS, L-BFGS, Keith O'Hara, Economics, Econometrics, Research, NYU, New York University" />
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<title>OptimLib: L-BFGS</title>
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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">OptimLib: L-BFGS</strong></h4></a>
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<a href="#definition">Definition</a> <br>
<a href="#details">Details</a> <br>
<a href="#examples">Examples</a>
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<p>The limited memeory BFGS (L-BFGS) algorithm is a quasi-Newton method for convex optimization.</p>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="definition"><strong style="font-size: 100%;">Definition and Syntax</strong></h3>
<!-- <br> -->
<pre class="brush: cpp;">
bool lbfgs(arma::vec& init_out_vals, std::function<double (const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)> opt_objfn, void* opt_data);
bool lbfgs(arma::vec& init_out_vals, std::function<double (const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)> opt_objfn, void* opt_data, algo_settings_t& settings);
</pre>
<p><strong>Function arguments:</strong></p>
<ul>
<li><code>init_out_vals</code> a column vector of initial values; will be replaced by the solution values.</li>
<li><code>opt_objfn</code> the function to be minimized, taking three arguments:
<ul>
<li><code>vals_inp</code> a vector of inputs;</li>
<li><code>grad_out</code> a vector to store the gradient; and</li>
<li><code>opt_data</code> additional parameters passed to the function.</li>
</ul>
<li><code>opt_data</code> additional parameters passed to the function.</li>
<li><code>value_out</code> the value of <code>opt_objfn</code> when evaulated at the output values.</li>
<li><code>settings</code> parameters controlling the optimization routine; see below.</li>
</ul>
<p><strong>Optimization <a href="optimlib_docs_settings.html" target="_blank">control parameters</a>:</strong></p>
<ul>
<li><code>double err_tol</code> the tolerance value controlling how small $\| \nabla f \|$ should be before 'convergence' is declared.</li>
<li><code>int iter_max</code> the maximum number of iterations/updates before the algorithm exits.</li>
<li><code>bool vals_bound</code> whether the search space is bounded. If true, then</li>
<ul>
<li><code>arma::vec lower_bounds</code> this defines the lower bounds.</li>
<li><code>arma::vec upper_bounds</code> this defines the upper bounds.</li>
</ul>
<br>
<li><code>int lbfgs_par_M</code> storage parameter.</li>
</ul>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="details"><strong style="font-size: 100%;">Details</strong></h3>
<p>Let $x^{(i)}$ denote the function input values at stage $i$ of the algorithm. The L-BFGS updating rule is as follows.</p>
<ul>
<li>The descent direction $d^{(i)}$ is found via a two-loop recursion. Let
\begin{align*}
s^{(i)} &:= x^{(i+1)} - x^{(i)} \\
y^{(i)} &:= \nabla f^{(i+1)} - \nabla f^{(i)}
\end{align*}
where $\nabla f^{(i)}$ is the gradient of $f$ evaluated at $x^{(i)}$.
<br>
Then:
<ul>
<li>Set $$q = \nabla f(x^{(i)})$$</li>
<li><strong>for</strong> $k = i - 1, \ldots, i - M$ <strong>do</strong>
\begin{align*}
\rho_k &= 1 / ( y^{(k)}{'} s^{(k)}) \\
\gamma_k &= \rho_k \times s^{(k)}{'} q \\
q &= q - \gamma_k \times y^{(k)}
\end{align*}
where the history length $M$ is set via <code>lbfgs_par_M</code>.</li>
<li>Set $r = q \times (s^{(i)}{'}y^{(i)} / y^{(i)}{'}y^{(i)})$</li>
<li><strong>for</strong> $k = i - M, \ldots, i - 1$ <strong>do</strong></li>
$$\beta = \rho_k y^{(k)}{'} r$$
$$r = r + s^{(k)}(\gamma_k - \beta)$$
<li>Finally, set $d^{(i)} = - r$.</li>
</ul>
</li>
<li>The input values are then updated:
$$x^{(i+1)} = x^{(i)} + \alpha^{(i)} d^{(i)}$$
where $\alpha^{(i)}$ is found via line search:
$$\alpha^{(i)} = \arg \min_\alpha f(x^{(i)} + \alpha d^{(i)})$$
The line search method used in OptimLib is that of More and Thuente (1994).</li>
</ul>
<p>The algorithm stops when $\| \nabla f \|$ is less than <code>err_tol</code>, or the total number of iterations exceeds <code>iter_max</code>.</p>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="examples"><strong style="font-size: 100%;">Examples</strong></h3>
<p>As examples, consider minimization of:</p>
<ul>
<li>the Sphere function
$$\min_{x \in \mathbb{R}^n} \sum_{i=1}^n x_i^2$$
The solution is the zero vector</li>
<li>Booth's function
$$\min_{x \in [-10,10]^2} \left\{ (x_1 + 2x_2 - 7)^2 + (2 x_1 + x_2 - 5)^2 \right\}$$
The global minimum is located at $(1,3)$.</li>
</ul>
<p>Below are plots of each example, restricted to the two-dimensional problem where relevant.</p>
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<h3 style="text-align: center;"><strong style="font-size: 110%;">Sphere</strong></h3>
<!-- <img class="img-responsive" src="pics/wsp_2.jpg" alt=""> -->
<img class="img-responsive center-block" src="pics/sphere_fn.png" alt="">
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<h3 style="text-align: center;"><strong style="font-size: 110%;">Booth</strong></h3>
<!-- <img class="img-responsive" src="pics/wsp_2.jpg" alt=""> -->
<!-- <img src="pics/rastrigin_fn.png" class="img-rounded pull-right" alt="Sphere" width="304" height="236"> -->
<img class="img-responsive center-block" src="pics/booth_fn_3d.png" alt="">
</div>
</div>
<hr>
<p>Code solving each example using OptimLib is given below.</p>
<pre class="brush: cpp;">
#include "optim.hpp"
double sphere_fn(const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)
{
double obj_val = arma::dot(vals_inp,vals_inp);
//
if (grad_out) {
*grad_out = 2.0*vals_inp;
}
//
return obj_val;
}
double booth_fn(const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)
{
double x_1 = vals_inp(0);
double x_2 = vals_inp(1);
double obj_val = std::pow(x_1 + 2*x_2 - 7.0,2) + std::pow(2*x_1 + x_2 - 5.0,2);
//
if (grad_out) {
(*grad_out)(0) = 2*(x_1 + 2*x_2 - 7.0) + 2*(2*x_1 + x_2 - 5.0)*2;
(*grad_out)(1) = 2*(x_1 + 2*x_2 - 7.0)*2 + 2*(2*x_1 + x_2 - 5.0);
}
//
return obj_val;
}
int main()
{
//
// sphere function
const int test_dim = 5;
arma::vec x = arma::ones(test_dim,1); // initial values (1,1,...,1)
bool success = optim::lbfgs(x,sphere_fn,nullptr);
if (success) {
std::cout << "lbfgs: sphere test completed successfully." << std::endl;
} else {
std::cout << "lbfgs: sphere test completed unsuccessfully." << std::endl;
}
arma::cout << "lbfgs: solution to sphere test:\n" << x << arma::endl;
//
// Booth function
arma::vec x_2 = arma::zeros(2,1); // initial values (0,0)
bool success_2 = optim::lbfgs(x,booth_fn,nullptr);
if (success_2) {
std::cout << "lbfgs: Booth test completed successfully." << std::endl;
} else {
std::cout << "lbfgs: Booth test completed unsuccessfully." << std::endl;
}
arma::cout << "lbfgs: solution to Booth test:\n" << x_2 << arma::endl;
return 0;
}
</pre>
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