  Stochastic optimization finds a wide range of applications in operations
research and management science. However, existing stochastic optimization
techniques usually require the information of random samples (e.g., demands in
the newsvendor problem) or the objective values at the sampled points (e.g.,
the lost sales cost), which might not be available in practice. In this paper,
we consider a new setup for stochastic optimization, in which the decision
maker has access to only comparative information between a random sample and
two chosen decision points in each iteration. We propose a comparison-based
algorithm (CBA) to solve such problems in one dimension with convex objective
functions. Particularly, the CBA properly chooses the two points in each
iteration and constructs an unbiased gradient estimate for the original
problem. We show that the CBA achieves the same convergence rate as the optimal
stochastic gradient methods (with the samples observed). We also consider
extensions of our approach to multi-dimensional quadratic problems as well as
problems with non-convex objective functions. Numerical experiments show that
the CBA performs well in test problems.
