Existence of Fixed-Point and Convergence Analysis of Piece-Wise Wide-Sense Standard Power Update Functions

Many power update functions proposed in the literature are functions of the effective interference (the ratio of the interference to the path gain). It is well known that for an upper-bounded wide-sense standard power update function (which is either standard or type-II standard function for each user), there exist a unique fixed-point and its convergence to its unique fixed-point is guaranteed. However, depending on the corresponding objective function, there exist power update functions which do not have the properties of monotonicity and scalability for all values of the effective interference (i.e., they are not wide-sense standard). For such a non-wide-sense standard power update function, the value-region of the effective interference for each user may be divided into several sub-regions, for each of which, a different wide-sense standard power update strategy may be employed, which we call it as a piece-wise wide-sense standard function. In this letter, we show that a bounded piece-wise wide-sense standard function is two-sided scalable and thus it poses a unique fixed-point and its convergence to its corresponding fixed-point is guaranteed.