  This paper considers a discrete-time decision problem wherein a decision
maker has to track, on average, a sequence of inputs selected from a convex set
$\mathcal X \subset \mathbb{R}^d$ by choosing actions from a possibly
non-convex feasible set $\mathcal Y\subset \mathbb{R}^d$, where $\mathcal X$ is
in fact the convex hull of $\mathcal Y$. We study some generalized variants of
this problem, in which: (i) $\mathcal X$ and $\mathcal Y$ vary with time, and
(ii) there might be a delay between them, in the sense that $\mathcal X$ is the
convex hull of the previous $\mathcal Y$. We investigate the conditions under
which the greedy algorithm that minimizes, in an online fashion, the
accumulated error between the sequence of inputs and decisions, is able to
track the average input asymptotically. Essentially, this comes down to proving
that the accumulated error, whose evolution is governed by a non-linear
dynamical system, remains within a bounded invariant set. Applications include
control of discrete devices using continuous setpoints; control of highly
uncertain devices with some information delay; and digital printing,
scheduling, and assignment problems.
