Regrets of proximal method of multipliers for online non-convex optimization with long term constraints

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