Strict complementarity in semidefinite programming, singularity degree, and the (dis)connection of forward and backward errors

Abstract

Strict complementarity of a primal-dual pair of optimal solutions is fundamental in the numerical analysis of semidefinite programs (SDPs). Strict complementarity drives the convergence behavior of interior point algorithms. When it fails, pathological examples show a striking gap between two error measures of approximate solutions. The first of these is the forward or "true" error, i.e., the distance to the optimal solution set. The second is the less useful backward error, measured by the constraint violation. We first characterize the lack of strict complementarity in SDPs via a simple normal form. Our normal form has three key features: (i) it is obtained using elementary row operations and rotations; (ii) it makes the lack of strict complementarity evident; and (iii) it lets us construct any such SDP by a simple algorithm. A variant of our generating algorithm allows us to construct any SDP that fails strict complementarity but satisfies Slater's condition on both the primal and dual sides. Thus, we {\em parametrize} the data of all SDPs that lack strict complementarity in a manner similar to how the Jordan normal form parametrizes square matrices with given eigenvalue structure. Next, we precisely characterize when the singularity degree of an SDP equals the number of constraints -- a result that underlies our generating algorithms and that we believe is of independent interest. We construct and share a set of SDPs that lack strict complementarity and present a detailed computational study. We find that forward-backward error gaps are quite common: in many small SDPs (with matrix order $\leq 20$), the forward ("true") error exceeds the backward error by up to seven orders of magnitude. Further, in several data sets the forward and backward errors are {\em inversely} correlated. In other words, the worse the "true" forward error is, the harder it is to detect.