.. _responses-objective_functions:

"""""""""""""""""""
objective_functions
"""""""""""""""""""


Response type suitable for optimization


.. toctree::
   :hidden:
   :maxdepth: 1

   responses-objective_functions-sense
   responses-objective_functions-primary_scale_types
   responses-objective_functions-primary_scales
   responses-objective_functions-weights
   responses-objective_functions-nonlinear_inequality_constraints
   responses-objective_functions-nonlinear_equality_constraints
   responses-objective_functions-scalar_objectives
   responses-objective_functions-field_objectives


**Specification**

- *Alias:* num_objective_functions 

- *Arguments:* INTEGER


**Child Keywords:**

+-------------------------+--------------------+--------------------------------------+---------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword                       | Dakota Keyword Description                  |
|                         | Group              |                                      |                                             |
+=========================+====================+======================================+=============================================+
| Optional                                     | `sense`__                            | Whether to minimize or maximize each        |
|                                              |                                      | objective function                          |
+----------------------------------------------+--------------------------------------+---------------------------------------------+
| Optional                                     | `primary_scale_types`__              | How to scale each objective function        |
+----------------------------------------------+--------------------------------------+---------------------------------------------+
| Optional                                     | `primary_scales`__                   | Characteristic values to scale each         |
|                                              |                                      | objective function                          |
+----------------------------------------------+--------------------------------------+---------------------------------------------+
| Optional                                     | `weights`__                          | Specify weights for each objective function |
+----------------------------------------------+--------------------------------------+---------------------------------------------+
| Optional                                     | `nonlinear_inequality_constraints`__ | Group to specify nonlinear inequality       |
|                                              |                                      | constraints                                 |
+----------------------------------------------+--------------------------------------+---------------------------------------------+
| Optional                                     | `nonlinear_equality_constraints`__   | Group to specify nonlinear equality         |
|                                              |                                      | constraints                                 |
+----------------------------------------------+--------------------------------------+---------------------------------------------+
| Optional                                     | `scalar_objectives`__                | Number of scalar objective functions        |
+----------------------------------------------+--------------------------------------+---------------------------------------------+
| Optional                                     | `field_objectives`__                 | Number of field objective functions         |
+----------------------------------------------+--------------------------------------+---------------------------------------------+

.. __: responses-objective_functions-sense.html
__ responses-objective_functions-primary_scale_types.html
__ responses-objective_functions-primary_scales.html
__ responses-objective_functions-weights.html
__ responses-objective_functions-nonlinear_inequality_constraints.html
__ responses-objective_functions-nonlinear_equality_constraints.html
__ responses-objective_functions-scalar_objectives.html
__ responses-objective_functions-field_objectives.html



**Description**


Specifies the number (1 or more) of objective functions :math:`f_j`
returned to Dakota for use in the general optimization problem
formulation:

.. math::

   \begin{eqnarray*}
   \hbox{minimize:} & & f(\mathbf{x}) = \sum_j{w_j f_j} \\
   & & \mathbf{x} \in \Re^{n} \\
   \hbox{subject to:} & &
   \mathbf{g}_{L} \leq \mathbf{g(x)} \leq \mathbf{g}_U \\
   & & \mathbf{h(x)}=\mathbf{h}_{t} \\
   & & \mathbf{a}_{L} \leq \mathbf{A}_i\mathbf{x} \leq \mathbf{a}_U \\
   & & \mathbf{A}_{e}\mathbf{x}=\mathbf{a}_{t} \\
   & & \mathbf{x}_{L} \leq \mathbf{x} \leq \mathbf{x}_U
   \end{eqnarray*}

Unless :dakkw:`responses-objective_functions-sense` is specified, Dakota
will minimize the objective functions.

The keywords
:dakkw:`responses-objective_functions-nonlinear_inequality_constraints` and
:dakkw:`responses-objective_functions-nonlinear_equality_constraints`
specify the number of nonlinear inequality constraints *g*, and nonlinear
equality constraints *h*, respectively.  When interfacing to external
applications, the responses must be returned to Dakota in this order
in the :dakkw:`interface-analysis_drivers-fork-results_file` :


1. objective functions
2. nonlinear_inequality_constraints
3. nonlinear_equality_constraints


An optimization problem's linear constraints are provided to the
solver at startup only and do not need to be included in the data
returned on every function evaluation. Linear constraints are
therefore specified in the :dakkw:`variables` block through the
:dakkw:`variables-linear_inequality_constraint_matrix` :math:`A_i`  and
:dakkw:`variables-linear_equality_constraint_matrix` :math:`A_e` .

Lower and upper bounds on the design variables *x* are also
specified in the :dakkw:`variables` block.

The optional keywords relate to scaling the objective functions (for
better numerical results), formulating the problem as minimization or
maximization, and dealing with multiple objective functions through
:dakkw:`responses-objective_functions-weights` *w*. If scaling is used,
it is applied before multi-objective weighted sums are formed, so,
e.g, when both weighting and characteristic value scaling are present
the ultimate objective function would be:


.. math::  f = \sum_{j=1}^{n} w_{j} \frac{ f_{j} }{ s_j }