We discuss here the convergence of relaxation methods for MPCC with approximate sequence of stationary points by presenting a general framework to study these methods. It has been pointed out in the literature, [25], that relaxation methods with approximate stationary points fail to give guarantee of convergence. We show that by de ning a new strong approximate stationarity we can attain the de...

We show that a waveguide with multiple gratings can have a modal dispersion relation which supports frozen light. This means that light can be coupled efficiently to low group velocity modes of an optical waveguide or can even have finite coupling to zero group velocity modes. These effects are associated with stationary points in the dispersion of the form ω − ωo ∝ (k − ko), for integer order ...

We present some new theoretical and computational results for the stationary points of bulk systems. First we demonstrate how the potential energy surface can be partitioned into catchment basins associated with every stationary point using a combination of Newton-Raphson and eigenvector-following techniques. Numerical results are presented for a 256-atom supercell representation of a binary Le...

This paper presents an asymptotic analysis of a Monte Carlo method, variously known as sample average approximation (SAA) or sample path optimization (SPO), for a general two-stage stochastic minimization problem. We study the case when the second-stage problem may have multiple local optima or stationary points that are not global solutions and SAA is implemented using a general nonlinear prog...

Optimality functions pioneered by E. Polak characterize stationary points, quantify the degree with which a point fails to be stationary, and play central roles in algorithm development. For optimization problems requiring approximations, optimality functions can be used to ensure consistency in approximations, with the consequence that optimal and stationary points of the approximate problems ...