Majorization as a tool for optimizing a class of matrix functions

The problem of minimizing a general matrix, trace function, possibly subject to certain constraints, is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found. It is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function, which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively. Three algorithms based on majorization for solving certain least squares problems are shown to be special cases. In addition, by means of several examples, it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available.The Netherlands organization for scientific research(NWO) is gratefully acknowledged for funding this project. This research was conducted while the author was supported by a PSYCHON-grant (560-267-011) from this organization. The author is obliged to Jos ten Berge, Willem Heiser, and Wim Krijnen for helpful comments on an earlier version of this paper.