Assume an objective function $f = g (X_{t_1},Y_{t_1}) + g (X_{t_2},Y_{t_2}) + g (X_{t_3},Y_{t_3}) + \dots+g (X_{t_n},Y_{t_n})$ where $g$ is a convex function. $X_{t_i}$ and $Y_{t_i}$ are sets of real numbers where $1 \le i \le n$ and the data points in $X_{t_i}$ and $Y_{t_i}$ arrives sequentially from a data stream. Hence, at time $t-1$, the data in $X_{t}$ and $Y_{t}$ is not available.

Assume the current time-stamp is $t$ and the window size is $s$, the goal is to minimize a series of object functions $f_t = g(X_{t},Y_{t}) + g(X_{t-1},Y_{t-1}) + \dots + g(X_{t-s+1},Y_{t-s+1})$ at time-stamp $t$, and compute the optimal sets of values $\bar{X}_{[t-s+1,t]}$ and $\bar{Y}_{[t-s+1,t]}$ to minimize the function. The data of $X$ and $Y$ arrived prior to time-stamp $t-s+1$ should be discarded and not considered in the scope in the minimization problem.

If we are at time-stamp $t+1$, window size is $s$ and minimization at time-stamp $t$ is done, data of $X_{t-s+1}$ and $Y_{t-s+1}$ should be discarded, but data of $X_{[t-s+2,t]}$ and $Y_{[t-s+2,t]}$ still needs to be used to minimize the function at time-stamp $t+1$. Is there a model to minimize function $f_{t+1}$ by using the data $X_{t+1}$ and $Y_{t+1}$ arrived at $t+1$ only without reprocessing the historical data in $X_{[t-s+2,t]}$ and $Y_{[t-s+2,t]}$?

Update 1:

$X_t$ and $Y_t$ are not varying, they can be treated as observations at $t$, and there are new observations at time $t+1$ which are $X_{t+1}$ and $Y_{t+1}$. I have not come up with an application yet, it is just for optimization with dynamic data. In a dynamic environment, the data can be changed at different time-stamps. The goal here is to minimize a function $f_t$ at time-stamp $t$ over most recent observations $X_{[t-s+1,t]}$ and $_{[t-s+1,t]}$ where $s$ is the fixed window size, and the minimization over $f_{t+1}$, $f_{t+1}$, $\dots$ shall be conducted over time. One naive way to do this is to re-process the whole data of $X_{[t-s+1,t]}$ and $_{[t-s+1,t]}$. This method may not be efficient since data of $X_{[t-s+1,t-1]}$ and $Y_{[t-s+1,t-1]}$ is already processed when minimizing $f_{t-1}$. Is there a method to minimize $f_t$ without re-scanning or re-processing the data $X_{[t-s+1,t-1]}$ and $Y_{[t-s+1,t-1]}$ by using some information when processing the minimization of $f_{t-1}$?