Dakota Reference Manual
Version 6.9
Explore and Predict with Confidence

The twovariable version of the ``textbook'' test problem provides a nonlinearly constrained optimization test case. It is formulated as:
Contours of this test problem are illustrated in the next two figures.
For the textbook test problem, the unconstrained minimum occurs at . However, the inclusion of the constraints moves the minimum to . Equation textbookform presents the 2dimensional form of the textbook problem. An extended formulation is stated as
where is the number of design variables. The objective function is designed to accommodate an arbitrary number of design variables in order to allow flexible testing of a variety of data sets. Contour plots for the case have been shown previously.
For the optimization problem given in Equation tbe, the unconstrained solution
(num_nonlinear_inequality_constraints
set to zero) for two design variables is:
with
The solution for the optimization problem constrained by \ (num_nonlinear_inequality_constraints
set to one) is:
with
The solution for the optimization problem constrained by and \ (num_nonlinear_inequality_constraints
set to two) is:
with
Note that as constraints are added, the design freedom is restricted (the additional constraints are active at the solution) and an increase in the optimal objective function is observed.