Active set identification and rapid convergence for degenerate primal-dual problems

Primal-dual methods for solving convex optimization problems with functional constraints often exhibit a distinct two-stage behavior. Initially, they converge towards a solution at a sublinear rate. Then, after a certain point, the method identifies the set of active constraints and the convergence enters a faster local linear regime. Theory characterizing this phenomenon spans over three decades. However, most existing work only guarantees eventual identification of the active set and relies heavily on nondegeneracy conditions, such as strict complementarity, which often fail to hold in practice. We characterize mild conditions on the problem geometry and the algorithm under which this phenomenon provably occurs. Our guarantees are entirely nonasymptotic and, importantly, do not rely on strict complementarity. Our framework encompasses several widely-used algorithms, including the proximal point method, the primal-dual hybrid gradient method, the alternating direction method of multipliers, and the extragradient method.

💡 Research Summary

This paper investigates a striking two‑phase convergence pattern that is routinely observed in first‑order primal‑dual algorithms for convex optimization problems with functional inequality constraints. In the first phase, iterates converge to a solution at a sublinear rate (typically O(1/√k) or O(1/k)). After a finite number of iterations, the algorithm “identifies” the active set of constraints—meaning that all subsequent iterates share the same set of binding inequalities—and the convergence switches to a faster, locally linear regime.

The classical theory explaining this phenomenon relies heavily on a non‑degeneracy condition known as strict complementarity: each active constraint must have a strictly positive Lagrange multiplier at the limit point. In practice, however, many problems (including the simple two‑dimensional quadratic program illustrated in Figure 1) violate strict complementarity, and existing results either provide only asymptotic guarantees or require algorithmic modifications specifically designed for degenerate cases.

The authors propose a unified, non‑asymptotic framework that removes the strict complementarity requirement while still delivering explicit finite‑time bounds for both active‑set identification and subsequent linear convergence. The key ingredients are:

1. Algorithmic assumptions – (i) the dual updates always stay in the non‑negative orthant (a property satisfied by PDHG, ADMM, Extragradient, and the Proximal Point Method), and (ii) the algorithm generates a sequence ((z_k)) with saddle‑subgradient norms decaying at a sublinear rate, i.e., (| \xi_k| = O(1/\sqrt{k})).

2. Problem‑side assumption – the mapping (F(z)) (the saddle‑subdifferential of the Lagrangian) satisfies a metric subregularity condition on a set (D) that contains all iterates:
   \