Consider a generic LP problem: $$ \begin{aligned} \max \;& \sum_{j=1}^n r_j x_j \\ \text {s.t. } & \sum_{j=1}^n a_{i j} x_j \leq b_i, \quad i=1, \ldots, m \\ & 0 \leq x_j \leq 1, j=1, \ldots, n, \end{aligned} \qquad \Longleftrightarrow \qquad \begin{aligned} \max \; &\sum_{j=1}^n r_j x_j \\ \text{s.t. } & \sum_{j=1}^n x_j \cdot \boldsymbol{a}_j \leq \boldsymbol{b} \\ & 0 \leq x_j \leq 1, j=1, \ldots, n, \end{aligned} $$ In the online setting, the parameters of the linear program are revealed in an online fashion, and one needs to determine the value of decision variables sequentially. Specifically, at each time $t$, the coefficients $(r_t, \boldsymbol{a}_t)$ are revealed, and we need to decide the value $x_t$ instantly.
Given the history $\mathcal{H}_{t-1} = \{r_j, \boldsymbol{a}_j, x_j\}^{t-1}_{j=1}$, the decision of $x_t$ can be expressed as a policy function $$ x_t = \pi_t(r_t, \boldsymbol{a}_t, \mathcal{H}_{t-1}) $$ The decision variable $x_t$ must conform to the constraints $$ \begin{gathered} \sum_{j=1}^t x_j \cdot \boldsymbol{a}_j \leq \boldsymbol{b}_j \\ 0 \leq x_t \leq 1 \end{gathered} $$
Denote the offline optimal solution as $\boldsymbol{x}^\ast = (x_1^\ast, x_2^\ast, \dots, x_n^\ast)^T$, the offline and online objective value as $R_n^\ast$ and $R_n(\boldsymbol{\pi})$ $$ \begin{aligned} R_n^\ast & :=\sum_{j=1}^n r_j x_j^\ast \\ R_n(\pi) & :=\sum_{j=1}^n r_j x_j . \end{aligned} $$
Pratical Examples
Online Knapsack Problem
In such problems, a decision maker chooses a subset of items in an online fashion to maximize the total value without violating the constraints.