  The online optimization problem with non-convex loss functions over a closed
convex set, coupled with a set of inequality (possibly non-convex) constraints
is a challenging online learning problem. A proximal method of multipliers with
quadratic approximations (named as OPMM) is presented to solve this online
non-convex optimization with long term constraints.
  Regrets of the violation of Karush-Kuhn-Tucker conditions of OPMM for solving
online non-convex optimization problems are analyzed. Under mild conditions, it
is shown that this algorithm exhibits ${\cO}(T^{-1/8})$ Lagrangian gradient
violation regret, ${\cO}(T^{-1/8})$ constraint violation regret and
${\cO}(T^{-1/4})$ complementarity residual regret if parameters in the
algorithm are properly chosen, where $T$ denotes the number of time periods.
For the case that the objective is a convex quadratic function, we demonstrate
that the regret of the objective reduction can be established even the feasible
set is non-convex. For the case when the constraint functions are convex, if
the solution of the subproblem in OPMM is obtained by solving its dual, OPMM is
proved to be an implementable projection method for solving the online
non-convex optimization problem.
