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Majorization Theory and Applications
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
In this chapter we introduce a useful mathematical tool, namely Majorization Theory, and illustrate its applications in a variety of scenarios in signal processing and communication systems. Majorization is a partial ordering and precisely defines the vague notion that the components of a vector are “less spread out” or “more nearly equal” than the components of another vector. Functions that preserve the ordering of majorization are said to be Schur-convex or Schur-concave. Many problems arising in signal processing and communications involve comparing vector-valued strategies or solving optimization problems with vector- or matrix-valued variables. Majorization theory is a key tool that allows us to solve or simplify these problems.
A Sharper Computational Tool for
Published in Technometrics, 2023
Xiaoqian Liu, Eric C. Chi, Kenneth Lange
Under these conditions, the iterates enjoy the descent property as demonstrated by the relationsreflecting conditions (3) and (4). Ideally, the MM principle converts a hard optimization problem into a sequence of easier ones. The key to success is the construction of a tight majorization that can be easily minimized. In some problems it is possible to construct a sharp majorization within a limited class of majorizers. Figure 1 depicts a sharp quadratic majorization that is best among all quadratic majorizations that share the same tangency point. Sharp majorization accelerates the convergence of a derived MM algorithm (de Leeuw and Lange 2009). In practice, majorization can be done piecemeal by exploiting the convexity or concavity of the various terms comprising the objective.