  In this paper, we consider online convex optimization (OCO) with time-varying
loss and constraint functions. Specifically, the decision maker chooses
sequential decisions based only on past information, meantime the loss and
constraint functions are revealed over time. We first develop a class of
model-based augmented Lagrangian methods (MALM) for time-varying functional
constrained OCO (without feedback delay). Under standard assumptions, we
establish sublinear regret and sublinear constraint violation of MALM.
Furthermore, we extend MALM to deal with time-varying functional constrained
OCO with delayed feedback, in which the feedback information of loss and
constraint functions is revealed to decision maker with delays. Without
additional assumptions, we also establish sublinear regret and sublinear
constraint violation for the delayed version of MALM. Finally, numerical
results for several examples of constrained OCO including online network
resource allocation, online logistic regression and online quadratically
constrained quadratical program are presented to demonstrate the efficiency of
the proposed algorithms.
