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<li class="navelem"><a class="el" href="namespace_quant_lib.html">QuantLib</a></li><li class="navelem"><a class="el" href="class_quant_lib_1_1_particle_swarm_optimization.html">ParticleSwarmOptimization</a></li> </ul>
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<p><code>#include <ql/experimental/math/particleswarmoptimization.hpp></code></p>
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<img id="dynsection-0-trigger" src="closed.png" alt="+"/> Inheritance diagram for ParticleSwarmOptimization:</div>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_inertia.html">Inertia</a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Base inertia class used to alter the PSO state. <a href="class_quant_lib_1_1_particle_swarm_optimization_1_1_inertia.html#details">More...</a><br /></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_topology.html">Topology</a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Base topology class used to determine the personal and global best. <a href="class_quant_lib_1_1_particle_swarm_optimization_1_1_topology.html#details">More...</a><br /></td></tr>
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 </td><td class="memItemRight" valign="bottom"><b>ParticleSwarmOptimization</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> M, const ext::shared_ptr< <a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_topology.html">Topology</a> > &topology, const ext::shared_ptr< <a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_inertia.html">Inertia</a> > &inertia, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> c1=2.05, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> c2=2.05, unsigned long seed=<a class="el" href="class_quant_lib_1_1_singleton.html#ab7455b7e1235d292c444095842349291">SeedGenerator::instance</a>().get())</td></tr>
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 </td><td class="memItemRight" valign="bottom"><b>ParticleSwarmOptimization</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> M, const ext::shared_ptr< <a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_topology.html">Topology</a> > &topology, const ext::shared_ptr< <a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_inertia.html">Inertia</a> > &inertia, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> omega, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> c1, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> c2, unsigned long seed=<a class="el" href="class_quant_lib_1_1_singleton.html#ab7455b7e1235d292c444095842349291">SeedGenerator::instance</a>().get())</td></tr>
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void </td><td class="memItemRight" valign="bottom"><b>startState</b> (<a class="el" href="class_quant_lib_1_1_problem.html">Problem</a> &P, const <a class="el" href="class_quant_lib_1_1_end_criteria.html">EndCriteria</a> &endCriteria)</td></tr>
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EndCriteria::Type </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_particle_swarm_optimization.html#a47e7853ef94a6bb46570c19401474db7">minimize</a> (<a class="el" href="class_quant_lib_1_1_problem.html">Problem</a> &P, const <a class="el" href="class_quant_lib_1_1_end_criteria.html">EndCriteria</a> &endCriteria)</td></tr>
<tr class="memdesc:a47e7853ef94a6bb46570c19401474db7"><td class="mdescLeft"> </td><td class="mdescRight">minimize the optimization problem P <br /></td></tr>
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std::vector< <a class="el" href="class_quant_lib_1_1_array.html">Array</a> > </td><td class="memItemRight" valign="bottom"><b>X_</b></td></tr>
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std::vector< <a class="el" href="class_quant_lib_1_1_array.html">Array</a> > </td><td class="memItemRight" valign="bottom"><b>V_</b></td></tr>
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std::vector< <a class="el" href="class_quant_lib_1_1_array.html">Array</a> > </td><td class="memItemRight" valign="bottom"><b>pBX_</b></td></tr>
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std::vector< <a class="el" href="class_quant_lib_1_1_array.html">Array</a> > </td><td class="memItemRight" valign="bottom"><b>gBX_</b></td></tr>
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<a class="el" href="class_quant_lib_1_1_array.html">Array</a> </td><td class="memItemRight" valign="bottom"><b>pBF_</b></td></tr>
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<a class="el" href="class_quant_lib_1_1_array.html">Array</a> </td><td class="memItemRight" valign="bottom"><b>gBF_</b></td></tr>
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<a class="el" href="class_quant_lib_1_1_array.html">Array</a> </td><td class="memItemRight" valign="bottom"><b>lX_</b></td></tr>
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<a class="el" href="class_quant_lib_1_1_array.html">Array</a> </td><td class="memItemRight" valign="bottom"><b>uX_</b></td></tr>
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<tr class="memitem:addcb16767a0a0fc4d2c3a54cb798464d"><td class="memItemLeft" align="right" valign="top"><a id="addcb16767a0a0fc4d2c3a54cb798464d"></a>
<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> </td><td class="memItemRight" valign="bottom"><b>M_</b></td></tr>
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<tr class="memitem:a7f499082c4cc12301dec25999dae6edc"><td class="memItemLeft" align="right" valign="top"><a id="a7f499082c4cc12301dec25999dae6edc"></a>
<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> </td><td class="memItemRight" valign="bottom"><b>N_</b></td></tr>
<tr class="separator:a7f499082c4cc12301dec25999dae6edc"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a9bf4db84e15b04a24f0f6cd4d1e82e06"><td class="memItemLeft" align="right" valign="top"><a id="a9bf4db84e15b04a24f0f6cd4d1e82e06"></a>
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>c0_</b></td></tr>
<tr class="separator:a9bf4db84e15b04a24f0f6cd4d1e82e06"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aa6b41d729f8349a20aa7a5a497f22682"><td class="memItemLeft" align="right" valign="top"><a id="aa6b41d729f8349a20aa7a5a497f22682"></a>
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>c1_</b></td></tr>
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<tr class="memitem:a0fd459b63cda42f7b4f1e33a8afec2c5"><td class="memItemLeft" align="right" valign="top"><a id="a0fd459b63cda42f7b4f1e33a8afec2c5"></a>
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>c2_</b></td></tr>
<tr class="separator:a0fd459b63cda42f7b4f1e33a8afec2c5"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a897450e49b1c8dde68a3f84ebf321bc7"><td class="memItemLeft" align="right" valign="top"><a id="a897450e49b1c8dde68a3f84ebf321bc7"></a>
<a class="el" href="class_quant_lib_1_1_mersenne_twister_uniform_rng.html">MersenneTwisterUniformRng</a> </td><td class="memItemRight" valign="bottom"><b>rng_</b></td></tr>
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ext::shared_ptr< <a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_topology.html">Topology</a> > </td><td class="memItemRight" valign="bottom"><b>topology_</b></td></tr>
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ext::shared_ptr< <a class="el" href="class_quant_lib_1_1_particle_swarm_optimization_1_1_inertia.html">Inertia</a> > </td><td class="memItemRight" valign="bottom"><b>inertia_</b></td></tr>
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Friends</h2></td></tr>
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class </td><td class="memItemRight" valign="bottom"><b>Inertia</b></td></tr>
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class </td><td class="memItemRight" valign="bottom"><b>Topology</b></td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>The process is as follows: M individuals are used to explore the N-dimensional parameter space: \( X_{i}^k = (X_{i, 1}^k, X_{i, 2}^k, \ldots, X_{i, N}^k) \) is the kth-iteration for the ith-individual.</p>
<p>X is updated via the rule </p><p class="formulaDsp">
\[ X_{i, j}^{k+1} = X_{i, j}^k + V_{i, j}^{k+1} \]
</p>
<p> with V being the "velocity" that updates the position: </p><p class="formulaDsp">
\[ V_{i, j}^{k+1} = \chi\left(V_{i, j}^k + c_1 r_{i, j}^k (P_{i, j}^k - X_{i, j}^k) + c_2 R_{i, j}^k (G_{i, j}^k - X_{i, j}^k)\right) \]
</p>
<p> where c are constants, r and R are uniformly distributed random numbers in the range [0, 1], and \( P_{i, j} \) is the personal best parameter set for individual i up to iteration k \( G_{i, j} \) is the global best parameter set for the swarm up to iteration k. \( c_1 \) is the self recognition coefficient \( c_2 \) is the social recognition coefficient</p>
<p>This version is known as the PSO with constriction factor (PSO-Co). PSO with inertia factor (PSO-In) updates the velocity according to: </p><p class="formulaDsp">
\[ V_{i, j}^{k+1} = \omega V_{i, j}^k + \hat{c}_1 r_{i, j}^k (P_{i, j}^k - X_{i, j}^k) + \hat{c}_2 R_{i, j}^k (G_{i, j}^k - X_{i, j}^k) \]
</p>
<p> and is accessible from PSO-Co by setting \( \omega = \chi \), and \( \hat{c}_{1,2} = \chi c_{1,2} \).</p>
<p>These two versions of PSO are normally referred to as canonical PSO.</p>
<p>Convergence of PSO-Co is improved if \( \chi \) is chosen as \( \chi = \frac{2}{\vert 2-\phi-\sqrt{\phi^2 - 4\phi}\vert} \), with \( \phi = c_1 + c_2 \). Stable convergence is achieved if \( \phi >= 4 \). Clerc and Kennedy recommend \( c_1 = c_2 = 2.05 \) and \( \phi = 4.1 \).</p>
<p>Different topologies can be chosen for G, e.g. instead of it being the best of the swarm, it is the best of the nearest neighbours, or some other form.</p>
<p>In the canonical PSO, the inertia function is trivial. It is simply a constant (the inertia) multiplying the previous iteration's velocity. The value of the inertia constant determines the weight of a global search over local search. Like in the case of the topology, other possibilities for the inertia function are also possible, e.g. a function that interpolates between a high inertia at the beginning of the optimization (hence prioritizing a global search) and a low inertia towards the end of the optimization (hence prioritizing a local search).</p>
<p>The optimization stops either because the number of iterations has been reached or because the stationary function value limit has been reached. </p>
<dl class="section examples"><dt>Examples</dt><dd><a class="el" href="_global_optimizer_8cpp-example.html#_a21">GlobalOptimizer.cpp</a>.</dd>
</dl>
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