Almost-sure convergence of iterates and multipliers in stochastic
  sequential quadratic optimization
Frank E. Curtis Xin Jiang Qi Wang
Stochastic sequential quadratic optimization (SQP) methods for solving
continuous optimization problems with nonlinear equality constraints have
attracted attention recently, such as for solving large-scale data-fitting
problems subject to nonconvex constraints. However, for a recently proposed
subclass of such methods that is built on the popular stochastic-gradient
methodology from the unconstrained setting, convergence guarantees have been
limited to the asymptotic convergence of the expected value of a stationarity
measure to zero. This is in contrast to the unconstrained setting in which
almost-sure convergence guarantees (of the gradient of the objective to zero)
can be proved for stochastic-gradient-based methods. In this paper, new
almost-sure convergence guarantees for the primal iterates, Lagrange
multipliers, and stationarity measures generated by a stochastic SQP algorithm
in this subclass of methods are proved. It is shown that the error in the
Lagrange multipliers can be bounded by the distance of the primal iterate to a
primal stationary point plus the error in the latest stochastic gradient
estimate. It is further shown that, subject to certain assumptions, this latter
error can be made to vanish by employing a running average of the Lagrange
multipliers that are computed during the run of the algorithm. The results of
numerical experiments are provided to demonstrate the proved theoretical
guarantees.