A limiting analysis on regularization of singular SDP and its implication to infeasible interior-point algorithms

Abstract We consider primal-dual pairs of semidefinite programs and assume that they are singular, i.e., both primal dual either weakly feasible or infeasible. Under such circumstances, strong duality may break down the might have a nonzero gap. Nevertheless, there arbitrary small perturbations to problem data which would make them strongly thus zeroing In this paper, we conduct an asymptotic analysis optimal value as perturbation for regularization is driven zero. Specifically, fix two positive definite matrices, $$I_p$$ I p $$I_d$$ d , say, (typically identity matrices), regularize problems by shifting their associated affine space $$\eta I_p$$ η $$\varepsilon I_d$$ ε respectively, recover interior feasibility problems, where $$ numbers. Then analyze behavior regularized when reduced zero keeping ratio between constant. A key feature our no further assumptions compactness constraint qualifications ever made. It will be shown perturbed converges values original problems. Furthermore, limiting changes “monotonically” from function $$\theta θ if parametrize $$(\varepsilon \eta )$$ ( , ) )=t(\cos \theta \sin = t cos sin let $$t\rightarrow 0$$ → 0 . Finally, leads us relatively surprising consequence some representative infeasible interior-point algorithms SDP generate sequences converging number values, even in presence Though result more theoretical interest at point, it development can handle singular