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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.differential_evolution"></a><a class="link" href="differential_evolution.html" title="Differential Evolution">Differential Evolution</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.differential_evolution.h0"></a>
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<span class="phrase"><a name="math_toolkit.differential_evolution.synopsis"></a></span><a class="link" href="differential_evolution.html#math_toolkit.differential_evolution.synopsis">Synopsis</a>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">optimization</span><span class="special">/</span><span class="identifier">differential_evolution</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">optimization</span> <span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ArgumentContainer</span><span class="special">></span> <span class="keyword">struct</span> <span class="identifier">differential_evolution_parameters</span> <span class="special">{</span>
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<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">ArgumentContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
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<span class="identifier">ArgumentContainer</span> <span class="identifier">lower_bounds</span><span class="special">;</span>
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<span class="identifier">ArgumentContainer</span> <span class="identifier">upper_bounds</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">mutation_factor</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>(</span><span class="number">0.65</span><span class="special">);</span>
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<span class="keyword">double</span> <span class="identifier">crossover_probability</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
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<span class="comment">// Population in each generation:</span>
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<span class="identifier">size_t</span> <span class="identifier">NP</span> <span class="special">=</span> <span class="number">200</span><span class="special">;</span>
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<span class="identifier">size_t</span> <span class="identifier">max_generations</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
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<span class="identifier">size_t</span> <span class="identifier">threads</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">thread</span><span class="special">::</span><span class="identifier">hardware_concurrency</span><span class="special">();</span>
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<span class="identifier">ArgumentContainer</span> <span class="keyword">const</span> <span class="special">*</span> <span class="identifier">initial_guess</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">;</span>
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<span class="special">};</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ArgumentContainer</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Func</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">URBG</span><span class="special">></span>
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<span class="identifier">ArgumentContainer</span> <span class="identifier">differential_evolution</span><span class="special">(</span>
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<span class="keyword">const</span> <span class="identifier">Func</span> <span class="identifier">cost_function</span><span class="special">,</span> <span class="identifier">differential_evolution_parameters</span><span class="special"><</span><span class="identifier">ArgumentContainer</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span><span class="identifier">de_params</span><span class="special">,</span> <span class="identifier">URBG</span> <span class="special">&</span><span class="identifier">g</span><span class="special">,</span>
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||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">></span> <span class="identifier">target_value</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">>>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">atomic</span><span class="special"><</span><span class="keyword">bool</span><span class="special">></span> <span class="special">*</span><span class="identifier">cancellation</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">,</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">ArgumentContainer</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">>>></span> <span class="special">*</span><span class="identifier">queries</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">,</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">atomic</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">>></span> <span class="special">*</span><span class="identifier">current_minimum_cost</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
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<span class="special">}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The <code class="computeroutput"><span class="identifier">differential_evolution</span></code>
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function provides an implementation of the (classical) differential evolution
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optimization algorithm, often going by the label <code class="computeroutput"><span class="identifier">de</span><span class="special">/</span><span class="identifier">rand</span><span class="special">/</span><span class="identifier">bin</span><span class="special">/</span><span class="number">1</span></code>.
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||
It is capable of minimizing a cost function defined on a continuous space represented
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by a set of bounds. This function has been designed more for progress observability,
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graceful cancellation, and post-hoc data analysis than for speed of convergence.
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||
We justify this design choice by reference to the "No free lunch"
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||
theorem of Wolpert and Macready, which establishes "that for any algorithm,
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any elevated performance over one class of problems is offset by performance
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over another class".
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||
</p>
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<h4>
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||
<a name="math_toolkit.differential_evolution.h1"></a>
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<span class="phrase"><a name="math_toolkit.differential_evolution.parameters"></a></span><a class="link" href="differential_evolution.html#math_toolkit.differential_evolution.parameters">Parameters</a>
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||
</h4>
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||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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||
<li class="listitem">
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<code class="computeroutput"><span class="identifier">lower_bounds</span></code>: A container
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||
representing the lower bounds of the optimization space along each dimension.
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The <code class="computeroutput"><span class="special">.</span><span class="identifier">size</span><span class="special">()</span></code> of the bounds should return the dimension
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of the problem.
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</li>
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<li class="listitem">
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<code class="computeroutput"><span class="identifier">upper_bounds</span></code>: A container
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representing the upper bounds of the optimization space along each dimension.
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||
It should have the same size of <code class="computeroutput"><span class="identifier">lower_bounds</span></code>,
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and each element should be >= the corresponding element of <code class="computeroutput"><span class="identifier">lower_bounds</span></code>.
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</li>
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<li class="listitem">
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||
<code class="computeroutput"><span class="identifier">mutation_factor</span></code>: Also known
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||
as <code class="computeroutput"><span class="identifier">F</span></code> or scale factor in
|
||
the literature. It controls the rate at which the population evolves (default
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||
is 0.65).
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</li>
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<li class="listitem">
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||
<code class="computeroutput"><span class="identifier">crossover_probability</span></code>:
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||
The crossover probability determining the trade-off between exploration
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and exploitation (default is 0.5).
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||
</li>
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||
<li class="listitem">
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||
<code class="computeroutput"><span class="identifier">NP</span></code>: The population size
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(default is 200). Parallelization occurs over the population, so this should
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||
be "large".
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||
</li>
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||
<li class="listitem">
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<code class="computeroutput"><span class="identifier">max_generations</span></code>: The maximum
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||
number of generations for the optimization (default is 1000).
|
||
</li>
|
||
<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">threads</span></code>: The number of
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||
threads to use for parallelization (default is the hardware concurrency).
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||
If the objective function is already multithreaded, then this should be
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||
set to 1 to prevent oversubscription.
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||
</li>
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||
<li class="listitem">
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||
<code class="computeroutput"><span class="identifier">initial_guess</span></code>: An optional
|
||
guess for where we should start looking for solutions.
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||
</li>
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||
</ul></div>
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<p>
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The defaults were chosen by a reading of Price, Storn, and Lampinen, chapter
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3, with the exception of the population size, which we have chosen a bit larger
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||
than they found as core counts have increased since publication of this reference
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||
and parallelization occurs within each generation. Note that there is a tradeoff
|
||
between finding global minima and convergence speed. The most robust way of
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||
decreasing the probability of getting stuck in a local minima is to increase
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||
the population size.
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||
</p>
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||
<h4>
|
||
<a name="math_toolkit.differential_evolution.h2"></a>
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||
<span class="phrase"><a name="math_toolkit.differential_evolution.the_function"></a></span><a class="link" href="differential_evolution.html#math_toolkit.differential_evolution.the_function">The
|
||
function</a>
|
||
</h4>
|
||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ArgumentContainer</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Func</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">URBG</span><span class="special">></span>
|
||
<span class="identifier">ArgumentContainer</span> <span class="identifier">differential_evolution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Func</span> <span class="identifier">cost_function</span><span class="special">,</span>
|
||
<span class="identifier">differential_evolution_parameters</span><span class="special"><</span><span class="identifier">ArgumentContainer</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span><span class="identifier">de_params</span><span class="special">,</span>
|
||
<span class="identifier">URBG</span> <span class="special">&</span><span class="identifier">gen</span><span class="special">,</span>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">></span> <span class="identifier">value_to_reach</span>
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||
<span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">>>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">atomic</span><span class="special"><</span><span class="keyword">bool</span><span class="special">></span> <span class="special">*</span><span class="identifier">cancellation</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">,</span>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">ArgumentContainer</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">>>></span> <span class="special">*</span><span class="identifier">queries</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">,</span>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">atomic</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">invoke_result_t</span><span class="special"><</span><span class="identifier">Func</span><span class="special">,</span> <span class="identifier">ArgumentContainer</span><span class="special">>></span> <span class="special">*</span><span class="identifier">current_minimum_cost</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">)</span>
|
||
</pre>
|
||
<p>
|
||
Parameters:
|
||
</p>
|
||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||
<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">cost_function</span></code>: The cost
|
||
function to be minimized.
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||
</li>
|
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<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">de_params</span></code>: The parameters
|
||
to the algorithm as described above.
|
||
</li>
|
||
<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">rng</span></code>: A uniform random bit
|
||
generator, like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937_64</span></code>.
|
||
</li>
|
||
<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">value_to_reach</span></code>: An optional
|
||
value that, if reached, stops the optimization. This is the most robust
|
||
way to terminate the calculation, but in most cases the optimal value of
|
||
the cost function is unknown. If it is, use it! See the referenced book
|
||
for clear examples of when target values can be deduced.
|
||
</li>
|
||
<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">cancellation</span></code>: An optional
|
||
atomic boolean to allow the user to stop the computation and gracefully
|
||
return the best result found up to that point. N.B.: Cancellation is not
|
||
immediate; the in-progress generation finishes.
|
||
</li>
|
||
<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">queries</span></code>: An optional vector
|
||
to store intermediate results during optimization. This is useful for debugging
|
||
and perhaps volume rendering of the objective function after the calculation
|
||
is complete.
|
||
</li>
|
||
<li class="listitem">
|
||
<code class="computeroutput"><span class="identifier">current_minimum_cost</span></code>: An
|
||
optional atomic variable to store the current minimum cost during optimization.
|
||
This allows developers to (e.g.) plot the progress of the minimization
|
||
over time and in conjunction with the cancellation argument allow halting
|
||
the computation when the progress stagnates. Refer to Price, Storn, and
|
||
Lampinen, Section 3.2 for caveats with this approach.
|
||
</li>
|
||
</ul></div>
|
||
<p>
|
||
Returns:
|
||
</p>
|
||
<p>
|
||
The <code class="computeroutput"><span class="identifier">ArgumentContainer</span></code> corresponding
|
||
to the minimum cost found by the optimization.
|
||
</p>
|
||
<p>
|
||
N.B.: The termination criteria is an "OR", not an "AND".
|
||
So if the maximum generations is hit, the iteration stops, even if (say) a
|
||
<code class="computeroutput"><span class="identifier">value_to_reach</span></code> has not been
|
||
attained.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.differential_evolution.h3"></a>
|
||
<span class="phrase"><a name="math_toolkit.differential_evolution.examples"></a></span><a class="link" href="differential_evolution.html#math_toolkit.differential_evolution.examples">Examples</a>
|
||
</h5>
|
||
<p>
|
||
An example use can be found <a href="../../../example/differential_evolution.cpp" target="_top">here</a>.
|
||
More examples and API use cases can be studied in <a href="../../../test/differential_evolution_test.cpp" target="_top">differential_evolution_test.cpp</a>.
|
||
</p>
|
||
<h6>
|
||
<a name="math_toolkit.differential_evolution.h4"></a>
|
||
<span class="phrase"><a name="math_toolkit.differential_evolution.caveats"></a></span><a class="link" href="differential_evolution.html#math_toolkit.differential_evolution.caveats">Caveats</a>
|
||
</h6>
|
||
<p>
|
||
We have decided to only support classic <code class="computeroutput"><span class="identifier">de</span><span class="special">/</span><span class="identifier">rand</span><span class="special">/</span><span class="number">1</span><span class="special">/</span><span class="identifier">bin</span></code>
|
||
because there are too many parameters for this class as it stands, and we have
|
||
not seen benchmarks that indicate that other variants of the algorithm perform
|
||
better. If a compelling usecase is provided, support for the <code class="computeroutput"><span class="identifier">de</span><span class="special">/</span><span class="identifier">x</span><span class="special">/</span><span class="identifier">y</span><span class="special">/</span><span class="identifier">z</span></code> variants can be added.
|
||
</p>
|
||
<p>
|
||
Supported termination criteria are explicit requests for termination, value-to-reach,
|
||
and max generations. Price, Storn, and Lampinen, Section 2.8 also list population
|
||
statistics and lack of accepted trials over many generations as sensible termination
|
||
criteria. These could be supported if there is demand.
|
||
</p>
|
||
<p>
|
||
Parallelization with <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">thread</span></code> does have overhead-especially for
|
||
very fast function calls. We found that the function call needs to be roughly
|
||
a microsecond for unambigous parallelization wins. However, we have not provided
|
||
conditional parallelization as computationally inexpensive cost functions are
|
||
the exception; not the rule. If there is a clear use case for conditional parallelization
|
||
(cheap cost function in very high dimensions is a good example), we can provide
|
||
it.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.differential_evolution.h5"></a>
|
||
<span class="phrase"><a name="math_toolkit.differential_evolution.references"></a></span><a class="link" href="differential_evolution.html#math_toolkit.differential_evolution.references">References</a>
|
||
</h5>
|
||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||
<li class="listitem">
|
||
Price, Kenneth, Rainer M. Storn, and Jouni A. Lampinen. <span class="emphasis"><em>Differential
|
||
evolution: a practical approach to global optimization.</em></span> Springer
|
||
Science & Business Media, 2006.
|
||
</li>
|
||
<li class="listitem">
|
||
D. H. Wolpert and W. G. Macready, <span class="emphasis"><em>No free lunch theorems for
|
||
optimization.</em></span> IEEE Transactions on Evolutionary Computation,
|
||
vol. 1, no. 1, pp. 67-82, April 1997, doi: 10.1109/4235.585893.
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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