We present a general approach to finding optimal arrangements of objects, given a cost function for evaluating an arrangement. The method is based on the assumption that features which are common to many weak local optima of a problem should be present in the global optimum. The algorithm identifies such common features and uses them to create “blocks” of objects which are treated as indivisible units. We have used general-purpose algorithms which do not exploit the peculiarities of any one problem. Thus, the method described here may not be as good as an heuristic which has been tailored to a particular problem. However, it is easily adaptable to different problems, and produces many near-optimal solutions.Two examples are discussed: an electrical-net wiring problem and the traveling salesman problem.

Finite-precision interval arithmetic evaluation of a function ƒ of n variables at an n -dimensional rectangle T which is the Cartesian product of intervals yields an interval which is denoted by F ( T ). Correspondingly, finite-precision real arithmetic evaluation of ƒ at the midpoint m ( T ) of T yields a number which is denoted by f ( m ( T )) Ε F ( T ). Often, f ( m ( T )) is surprisingly close to m ( F ( T )). The purpose of this note is to provide some insight into this phenomenon by examining the case of infinite precision and rational functions. It is shown that if the gradient of ƒ is nonzero at a fixed point t Ε T , then as the maximum edge length w ( T ) of T approaches zero, [ m ( F ( T )) - ƒ( m ( T ))]/ w ( F ( T )) = O ( w ( T )), where F ( T ) and ƒ( m ( T )) denote the infinite-precision results corresponding to F ( T ) and f ( m ( T )), respectively. More precise results are derived when ƒ is one of +, -, ×, or /.