The Computational Complexity of Avoiding Strict Saddle Points in Constrained Optimization
2026-04-02 • Computational Complexity
Computational ComplexityData Structures and Algorithms
AI summaryⓘ
The authors study the difficulty of finding second-order stationary points (SOSPs) in constrained non-convex optimization problems. They show that under a common, manageable definition of SOSPs, finding approximate solutions is PLS-complete, meaning it is computationally hard. This holds even in simple 2D settings and when solutions are far from the domain boundaries. Their work clarifies the complexity of constrained SOSP problems, contrasting with prior results for first-order points, and reveals limits on the existence of efficient, deterministic algorithms for this task.
non-convex optimizationfirst-order stationary pointssecond-order stationary pointsPLS-completenessconstrained optimizationKKT pointscomputational complexityapproximate stationary pointsPPADTFNP
Authors
Andreas Kontogiannis, Ioannis Panageas, Vasilis Pollatos
Abstract
While first-order stationary points (FOSPs) are the traditional targets of non-convex optimization, they often correspond to undesirable strict saddle points. To circumvent this, attention has shifted towards second-order stationary points (SOSPs). In unconstrained settings, finding approximate SOSPs is PLS-complete (Kontogiannis et al.), matching the complexity of finding unconstrained FOSPs (Hollender and Zampetakis). However, the complexity of finding SOSPs in constrained settings remained notoriously unclear and was highlighted as an important open question by both aforementioned works. Under one strict definition, even verifying whether a point is an approximate SOSP is NP-hard (Murty and Kabadi). Under another widely adopted, relaxed definition where non-negative curvature is required only along the null space of the active constraints, the problem lies in TFNP, and algorithms with O(poly(1/epsilon)) running times have been proposed (Lu et al.). In this work, we settle the complexity of constrained SOSP by proving that computing an epsilon-approximate SOSP under the tractable definition is PLS-complete. We demonstrate that our result holds even in the 2D unit square [0,1]^2, and remarkably, even when stationary points are isolated at a distance of Omega(1) from the domain's boundary. Our result establishes a fundamental barrier: unless PLS is a subset of PPAD (implying PLS = CLS), no deterministic, iterative algorithm with an efficient, continuous update rule can exist for finding approximate SOSPs. This contrasts with the constrained first-order counterpart, for which Fearnley et al. showed that finding an approximate KKT point is CLS-complete. Finally, our result yields the first problem defined in a compact domain to be shown PLS-complete beyond the canonical Real-LocalOpt (Daskalakis and Papadimitriou)."