Condensed Information about the Test Problems

Abstract

To give a first survey of the test examples and their solution properties, we present a comprehensive list of all problems in Table 2. Beside the current problem number and the classification number OCD-Kr-s as described in Section 3 of Chapter I, we report the dimension n, the number of all inequality constraints m1, the number of all equality constraints m-m1, and the number of all bounds b. If linear restrictions exist, their number is given in the brackets behind m1 or m-m1, respectively. The column headed by x0 gives the information, whether the starting point x0 is feasible (T) or not (F). Some numerical data obtained by the analysis described in Section 2 of Chapter I are encluded. In particular, the objective function value f(x*), the sum of constraint violations r(x*), the norm of the Kuhn-Tucker-vector e(x*), the number of active plus equality restrictions, i.e. \(\overline {\rm \mu } :\, = \,{\rm \mu }\,{\rm + }\,{\rm m}\,{\rm - }\,{\rm m}_{\rm 1} \), the degree of degeneracy \({\rm u}_{\max }^* /{\rm u}_{{\rm min}}^{\rm *} \), and the condition number of the Hessian of the projected Lagrangian \({\rm \lambda }_{{\rm max}}^{\rm *} /{\rm \lambda }_{{\rm min}}^{\rm *} \) are listed. A value of 0. for \({\rm u}_{\max }^* /{\rm u}_{{\rm min}}^{\rm *} \) indicates that at least one multiplier \({\rm u}_{\rm j}^{\rm *} \) vanishes (\(\left| {{\rm u}_{\rm j}^{\rm *} } \right|\, < \,.\,1{\rm E - 5}\)) and that there are redundant constraints, i.e. the problem is degenerate. By \({\rm \lambda }_{{\rm max}}^{\rm *} /{\rm \lambda }_{{\rm min}}^{\rm *} \).