Unconstrained global optimization using strict complementary slackness (Q811408)

The author considers the problem of minimizing a continuously differentiable function f over a given parallelepiped in \({\mathbb{R}}^ n\) with sides parallel to the coordinate axes. This problem is considered as a special constrained optimization problem, for which the first order constraint qualification always holds. A simple example is presented showing that the strict complementary slackness condition does not always hold at a Kuhn-Tucker point of the given problem. A necessary and sufficient condition for the strict complementary slackness is formulated in terms of the partial derivatives of f. The final section contains a rather unclear discussion of how to use the strict complementary slackness to bisect the parallelepiped for locating the local minimizers of the given problem.