Transformation of min-max optimization to least-square estimation and application to interconnect design optimization

This paper describes a new approach to find a tighter bound of the transformation of the min-max problems into the one of least-square estimation. It is well known that the above transformation of one problem to the other can lead to the proof that their target functions linearly bound each other. However, this linear bound is not a tight one. In this paper, we prove that if we transform the min-max problem into two least-square estimation problems, where one minimizes the root-mean-square (RMS) of the original function and the other one minimizes the RMS of the difference between the original function and an arbitrary constant, one can obtain a tighter bound between their target functions. The tighter bound given by this new approach depends on the outcome of the second least-square estimation problem, so there is a great incentive to choose the arbitrary constant which gives the smallest RMS of the second least-square estimation problem. For a problem with a large number of variables, this new tighter bound can be two to three order tighter than the old one.