  We study a competitive online optimization problem with multiple inventories.
In the problem, an online decision maker seeks to optimize the allocation of
multiple capacity-limited inventories over a slotted horizon, while the
allocation constraints and revenue function come online at each slot. The
problem is challenging as we need to allocate limited inventories under
adversarial revenue functions and allocation constraints, while our decisions
are coupled among multiple inventories and different slots. We propose a
divide-and-conquer approach that allows us to decompose the problem into
several single inventory problems and solve it in a two-step manner with almost
no optimality loss in terms of competitive ratio (CR). Our approach provides
new angles, insights and results to the problem, which differs from the
widely-adopted primal-and-dual framework. Specifically, when the gradients of
the revenue functions are bounded in a positive range, we show that our
approach can achieve a tight CR that is optimal when the number of inventories
is small, which is better than all existing ones. For an arbitrary number of
inventories, the CR we achieve is within an additive constant of one to a lower
bound of the best possible CR among all online algorithms for the problem. We
further characterize a general condition for generalizing our approach to
different applications. For example, for a generalized one-way trading problem
with price elasticity, where no previous results are available, our approach
obtains an online algorithm that achieves the optimal CR up to a constant
factor.
