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