  This paper studies the online stochastic resource allocation problem (RAP)
with chance constraints and conditional expectation constraints. The online RAP
is an integer linear programming problem where resource consumption
coefficients are revealed column by column along with the corresponding revenue
coefficients. When a column is revealed, the corresponding decision variables
are determined instantaneously without future information. In online
applications, the resource consumption coefficients are often obtained by
prediction. An application for such scenario rises from the online order
fulfilment task. When the timeliness constraints are considered, the
coefficients are generated by the prediction for the transportation time from
origin to destination. To model their uncertainties, we take the chance
constraints and conditional expectation constraints into the consideration.
Assuming that the uncertain variables have known Gaussian distributions, the
stochastic RAP can be transformed into a deterministic but nonlinear problem
with integer second-order cone constraints. Next, we linearize this nonlinear
problem and theoretically analyze the performance of vanilla online primal-dual
algorithm for solving the linearized stochastic RAP. Under mild technical
assumptions, the optimality gap and constraint violation are both on the order
of $\sqrt{n}$. Then, to further improve the performance of the algorithm,
several modified online primal-dual algorithms with heuristic corrections are
proposed. Finally, extensive numerical experiments demonstrate the
applicability and effectiveness of our methods.
