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This paper considers online convex optimization with time-varying constraintfunctions. Specifically, we have a sequence of convex objective functions$\{f_t(x)\}_{t=0}^{\infty}$ and convex constraint functions$\{g_{t,i}(x)\}_{t=0}^{\infty}$ for $i \in \{1, ..., k\}$. The functions aregradually revealed over time. For a given $\epsilon>0$, the goal is to choosepoints $x_t$ every step $t$, without knowing the $f_t$ and $g_{t,i}$ functionson that step, to achieve a time average at most $\epsilon$ worse than the bestfixed-decision that could be chosen with hindsight, subject to the time averageof the constraint functions being nonpositive. It is known that this goal isgenerally impossible. This paper develops an online algorithm that solves theproblem with $O(1/\epsilon^2)$ convergence time in the special case when allconstraint functions are nonpositive over a common subset of $\mathbb{R}^n$.Similar performance is shown in an expected sense when the common subsetassumption is removed but the constraint functions are assumed to varyaccording to a random process that is independent and identically distributed(i.i.d.) over time slots $t \in \{0, 1, 2, \ldots\}$. Finally, in the specialcase when both the constraint and objective functions are i.i.d. over timeslots $t$, the algorithm is shown to come within $\epsilon$ of optimality withrespect to the best (possibly time-varying) causal policy that knows the fullprobability distribution.

Online Convex Optimization with Time-Varying Constraints