.. _MultiMinimizationObjective:


Multi-objective minimization
----------------------------


The objective is to learn the *Pareto front* of multiple minimization objectives.  A vector of objective values :math:`\mathbf{y}\in \mathbb{R}^q` lies at the Pareto front if the improvement (decrease) if it is not dominated by any other possible vector of outcomes. A vector :math:`\mathbf{y}_1` is dominated by a vector :math:`\mathbf{y}_2` if all entries of :math:`\mathbf{y}_2` are smaller or equal the entries of :math:`\mathbf{y}_1`.  

The strategy to find samples close or at the Pareto front is to maximize the hypervolume enclosed by all non-dominated sampling points and the upper reference point :math:`\mathbf{y}_{\rm upper}`. The progress is defined in terms of the decreasing hypervolume between the lower reference point :math:`\mathbf{y}_{\rm lower}` and the non-dominated sampling points.

.. _ActiveLearning.MultiMinimizer.name:


name (str)
""""""""""
   The name of the objective. 

   Default: ``'objective'``

.. _ActiveLearning.MultiMinimizer.variables:


variables (list[str])
"""""""""""""""""""""
   A list with the names of the  variables that specify the objective. 

   Default: A list with the first two variables in the list of all  variables.

.. _ActiveLearning.MultiMinimizer.lower_reference:


lower_reference (list[float])
"""""""""""""""""""""""""""""
   Specifies the reference point :math:`\mathbf{y}_{\rm lower}` of the hypervolume. The entries should be at maximum slightly smaller than the minimal possible value of each objective. 

   Default: This value has no default and must be provided

.. _ActiveLearning.MultiMinimizer.upper_reference:


upper_reference (list[float])
"""""""""""""""""""""""""""""
   Specifies the reference point :math:`\mathbf{y}_{\rm upper}` of the hypervolume. The entries should be at least slightly larger than the maximal possible value of each objective. 

   Default: This value has no default and must be provided

.. _ActiveLearning.MultiMinimizer.partitioning_tolerance:


partitioning_tolerance (float)
""""""""""""""""""""""""""""""
   The parameter can be increased to obtain an approximate  partitioning faster. The tolerance is a fraction of the total hypervolume.  When a hypercell's volume divided by the total hypervolume is less than  the tolerance, it is disregarded. 

   Default: ``0.0``

.. _ActiveLearning.MultiMinimizer.localize:


localize (bool)
"""""""""""""""
   If true, a local search is performed, i.e.  samples are not drawn in regions with large uncertainty. 

   Default: ``False``

.. _ActiveLearning.MultiMinimizer.min_acq_val:


min_acq_val (float)
"""""""""""""""""""
   The study is stopped if the maximum acquisition value (usually expected hypervolume improvement) of the last 5 iterations is below ``min_acq_val``. 

   Default: ``1e-16``