Supercalm Multifunctions For Convergence Analysis

Abstract

Calmness of multifunctions is a well-studied concept of generalized continuity in which single-valued selections from the image sets of the multifunction exhibit a restricted type of local Lipschitz continuity where the base point is fixed as one point of comparison. Generalized continuity properties of multifunctions like calmness can be applied to convergence analysis when the multifunction appropriately represents the iterates generated by some algorithm. Since it involves an essentially linear relationship between input and output, calmness gives essentially linear convergence results when it is applied directly to convergence analysis. We introduce a new continuity concept called ‘supercalmness’ where arbitrarily small calmness constants can be obtained near the base point, which leads to essentially superlinear convergence results. We also explore partial supercalmness and use a well-known generalized derivative to characterize both when a multifunction is supercalm and when it is partially supercalm. To illustrate the value of such characterizations, we explore in detail a new example of a general primal sequential quadratic programming method for nonlinear programming and obtain verifiable conditions to ensure convergence at a superlinear rate.