Matrix optimization problems for MIMO systems with matrix monotone objective functions

In this paper, various optimization problems with matrix variates for multiple-input multiple-output (MIMO) systems are unified into a novel optimization framework namely matrix-monotonic optimization problems. Monotonicity is one of the most important and fundamental characteristics of a function, which can be exploited to derive and analyze the optimal solutions. In this paper, we discover that in several cases taking advantage of the monotonicity in the field of positive semidefinite matrices, the considered optimization problems can be significantly simplified e.g., the dimensionality of the variables are reduced from matrix variates to be vector ones with much lower dimensionality. We believe that just like convexity, monotonicity, especially for positive semi-definite matrices, will have a critical role in the future wireless designs.