  We propose the algorithms for online convex optimization which lead to
cumulative squared constraint violations of the form
$\sum\limits_{t=1}^T\big([g(x_t)]_+\big)^2=O(T^{1-\beta})$, where
$\beta\in(0,1)$. Previous literature has focused on long-term constraints of
the form $\sum\limits_{t=1}^Tg(x_t)$. There, strictly feasible solutions can
cancel out the effects of violated constraints. In contrast, the new form
heavily penalizes large constraint violations and cancellation effects cannot
occur.
  Furthermore, useful bounds on the single step constraint violation
$[g(x_t)]_+$ are derived.
  For convex objectives, our regret bounds generalize existing bounds, and for
strongly convex objectives we give improved regret bounds.
  In numerical experiments, we show that our algorithm closely follows the
constraint boundary leading to low cumulative violation.
