Level-set methods for convex optimization are predicated on the idea that certain problems can be parameterized so that their solutions can be recovered as the limiting process of a root-finding procedure. This idea emerges time and again across a range of algorithms for convex problems. Here we demonstrate that strong duality is a necessary condition for the level-set approach to succeed. In the absence of strong duality, the level-set method identifies \(\epsilon \)-infeasible points that do not converge to a feasible point as \(\epsilon \) tends to zero. The level-set approach is also used as a proof technique for establishing sufficient conditions for strong duality that are different from Slater’s constraint qualification.

凸优化的水平集方法基于以下思想：可以对某些问题进行参数化，以便可以将其解决方案作为寻根过程的限制过程来恢复。这个想法在用于凸问题的一系列算法中一次又一次地出现。在这里，我们证明了强大的对偶性是成功设定水平方法的必要条件。在没有强对偶的情况下，水平集方法会识别\（\ epsilon \）-不可行点，因为\（\ epsilon \）趋于零，这些点不会收敛到可行点。水平集方法还可以用作证明技术，以建立不同于Slater约束条件的强对偶性条件。