Dakota Reference Manual: Textbook

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The two-variable 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.

Contours of the textbook problem on the [-3,4] x [-3,4] domain. The feasible region lies at the intersection of the two constraints g_1 (solid) and g_2 (dashed).
Contours of the textbook problem zoomed into an area containing the constrained optimum point (x_1,x_2) = (0.5,0.5). The feasible region lies at the intersection of the two constraints g_1 (solid) and g_2 (dashed).

For the textbook test problem, the unconstrained minimum occurs at . However, the inclusion of the constraints moves the minimum to . Equation textbookform presents the 2-dimensional 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.