This proposition can be proved by dividing the unimodal function u 0(r) into two types of component ( {u}_0^1(r) ) and ( {u}_0^2(r) ) with the splitpoint ( frac{r_1+{r}{2}{2} ), where ( {u}_0^1(r) ) and ( {u}_0^2(r) ) are monotonically increasing and decreasing step functions, respectively, and then using the conclusion in Proposition 1. Example 3.
This implies that the sequence is monotonically increasing, provided that the step sizes are sufficiently small.
Instead of studying those particular polyhedra, we consider a much more general object, the unitary step monotonically increasing set function polyhedra, and identify some families of facets.
The results, show that unlike in the stationary case, the steady-state EMSE is not a monotonically increasing function of the step-size µ, while the ability of the algorithm to track the variations in the environment degrades by increasing the frequency offset Ω.
The iteration between E- and M-steps gives monotonically increasing series of the values for the log-likelihood.
Step 2. There exists a monotonically increasing sequence ({m_{k}} ) such that the sequence (bar{delta}_{m_{k}}) is increasing on (m_{k}).
Step 2. There exists a monotonically increasing sequence ({m_{k}} ) such that the sequence ({underline{delta}_{m_{k}}}) is decreasing on (m_{k}) and ({overline{delta}_{m_{k}}}) is increasing on (m_{k}).
Consequently, the D surface coverage is governed by the kinetics of this elementary step, thereby resulting in a monotonically increasing surface coverage with increasing temperature.
The sensitivity analysis on the service level in a partial line-haul operations network for the second largest carrier in Taiwan showed that the costs are not strictly monotonically increasing with the service levels, rather they are monotonically non-decreasing according to a step function.
Although there can be large changes in the task parameters from one step to the next, there is a guarantee that the average reward is monotonically increasing.
