On the conditional acceptance of iterates in SAO algorithms based on convex separable approximations

We reflect on the convergence and termination of optimization algorithms based on convex and separable approximations using two recently proposed strategies, namely a trust region with filtered acceptance of the iterates, and conservatism. We then propose a new strategy for convergence and termination, denoted filtered conservatism, in which the acceptance or rejection of an iterate is determined using the nonlinear acceptance filter. However, if an iterate is rejected, we increase the conservatism of every unconservative approximation, rather than reducing the trust region. Filtered conservatism aims to combine the salient features of trust region strategies with nonlinear acceptance filters on the one hand, and conservatism on the other. In filtered conservatism, the nonlinear acceptance filter is used to decide if an iterate is accepted or rejected. This allows for the acceptance of infeasible iterates, which would not be accepted in a method based on conservatism. If however an iterate is rejected, the trust region need not be decreased; it may be kept constant. Convergence is than effected by increasing the conservatism of only the unconservative approximations in the (large, constant) trust region, until the iterate becomes acceptable to the filter. Numerical results corroborate the accuracy and robustness of the method.