1Blurb::
2Response type suitable for optimization
3
4Description::
5
6Specifies the number (1 or more) of objective functions \f$ f_j \f$
7returned to Dakota for use in the general optimization problem
8formulation:
9
10\f{eqnarray*}
11  \hbox{minimize:} & & f(\mathbf{x}) = \sum_j{w_j f_j} \\
12  & & \mathbf{x} \in \Re^{n} \\
13  \hbox{subject to:} & &
14  \mathbf{g}_{L} \leq \mathbf{g(x)} \leq \mathbf{g}_U \\
15  & & \mathbf{h(x)}=\mathbf{h}_{t} \\
16  & & \mathbf{a}_{L} \leq \mathbf{A}_i\mathbf{x} \leq \mathbf{a}_U \\
17  & & \mathbf{A}_{e}\mathbf{x}=\mathbf{a}_{t} \\
18  & & \mathbf{x}_{L} \leq \mathbf{x} \leq \mathbf{x}_U
19\f}
20
21Unless \ref responses-objective_functions-sense is specified, Dakota
22will minimize the objective functions.
23
24The keywords \ref
25responses-objective_functions-nonlinear_inequality_constraints and
26\ref responses-objective_functions-nonlinear_equality_constraints
27specify the number of nonlinear inequality constraints \em g, and nonlinear
28equality constraints \em h, respectively.  When interfacing to external
29applications, the responses must be returned to %Dakota in this order
30in the \ref interface-analysis_drivers-fork-results_file :
31<ol>
32  <li>objective functions</li>
33  <li>nonlinear_inequality_constraints</li>
34  <li>nonlinear_equality_constraints</li>
35</ol>
36
37An optimization problem's linear constraints are provided to the
38solver at startup only and do not need to be included in the data
39returned on every function evaluation. Linear constraints are
40therefore specified in the \ref variables block through the \ref
41variables-linear_inequality_constraint_matrix \f$A_i\f$ and \ref
42variables-linear_equality_constraint_matrix \f$A_e\f$.
43
44Lower and upper bounds on the design variables \em x are also
45specified in the \ref variables block.
46
47The optional keywords relate to scaling the objective functions (for
48better numerical results), formulating the problem as minimization or
49maximization, and dealing with multiple objective functions through
50\ref responses-objective_functions-weights \em w.  If scaling is used,
51it is applied before multi-objective weighted sums are formed, so,
52e.g, when both weighting and characteristic value scaling are present
53the ultimate objective function would be:
54
55\f[ f = \sum_{j=1}^{n} w_{j} \frac{ f_{j} }{ s_j } \f]
56
57Topics::
58Examples::
59Theory::
60Faq::
61See_Also::	responses-calibration_terms, responses-response_functions, method, variables
62