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We consider primal-dual pairs of semidefinite programs and assume that theyare ill-posed, i.e., both primal and dual are either weakly feasible or weaklyinfeasible. Under such circumstances, strong duality may break down and theprimal and dual might have a nonzero duality gap. Nevertheless, there arearbitrary small perturbations to the problem data which makes the perturbedprimal-dual pair strongly feasible thus zeroing the duality gap. In this paper,we conduct an asymptotic analysis of the optimal value as the perturbation isdriven to zero. Specifically, we fix two positive definite matrices (typicallythe identity matrices), and shift the associated affine spaces of the primaland dual slightly in the direction of the two positive definite matricespossibly in a different proportion so that the perturbed problems have interiorfeasible solutions, and analyze the behavior of the optimal value of theperturbed problem when the perturbation is reduced to zero keeping theproportion. A key feature of our analysis is that no further assumptions suchas compactness or constraint qualifications are ever made. It will be shownthat the optimal value of the perturbed problem converges to a value betweenthe primal and dual optimal values of the original problem. Finally, theanalysis leads us to the relatively surprising consequence that the infeasibleinterior-point algorithms for SDP generates a sequence converging to a numberbetween the primal and dual optimal values, even in the presence of a nonzeroduality gap. We expect that this property might be particularly useful insolving mixed integer SDPs with infeasible interior-point methods.

A Limiting Analysis on Regularization of Singular SDP and its Implication to Infeasible Interior-point Algorithms