Dakota Reference Manual  Version 6.12
Explore and Predict with Confidence
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objective_functions


Response type suitable for optimization

Specification

Alias: num_objective_functions

Argument(s): INTEGER

Child Keywords:

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
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

Description

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

\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 sense is specified, Dakota will minimize the objective functions.

The keywords nonlinear_inequality_constraints and 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 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 variables block through the linear_inequality_constraint_matrix $A_i$ and linear_equality_constraint_matrix $A_e$.

Lower and upper bounds on the design variables x are also specified in the 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 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:

\[ f = \sum_{j=1}^{n} w_{j} \frac{ f_{j} }{ s_j } \]

See Also

These keywords may also be of interest: