Counterpart and Correction for Strong Duality of Second-Order Conic Program in Radial Networks

It was reported in the literature that the strong duality does not hold in general for the second-order conic program (SOCP) based AC optimal power flow in radial networks, which challenged many robust and stochastic methods using the dual form of SOCP. However, the letter shows that this viewpoint is wrong. In fact, SOCP does have a strong duality in general. Moreover, we give strict proof and counterpart examples to show the tightness of the strong duality. Finally, we investigate that the wrong conclusion in the literature is mainly resulted from a wrong duality form. The conclusion of this letter will support the utilizations of duality in SOCP based robust or stochastic optimization models.