In this paper, we study dual semismooth Newton (SSN) methods for degenerate polyhedral projection problems, where generalized Jacobians of the dual residual may remain singular even arbitrarily close to the solution set. Rather than regularizing these singular systems, we exploit the nonuniqueness of the dual representation. We introduce a primal--dual lifted projection-equivalent set that always possesses extreme points without additional structural assumptions on the polyhedron, and show that its extreme-point geometry identifies dual representatives at which nonsingular generalized Jacobians of the dual residual can be constructed. This geometry is further linked to a full-column-rank condition and a generalized weak strict Robinson constraint qualification, showing that the regularity required by the Newton step can be recovered rather than imposed \emph{a priori}. We also establish displacement bounds that connect representative selection throughout the algorithm with the local Newton mechanism. Building on this variational framework, we develop an inexact dual SSN method with local superlinear convergence and a globalized version combining monotone representative selection with a Wolfe line search. The resulting method is globally convergent and eventually recovers the fast local rate. Numerical experiments on regularized optimal transport, battery-scheduling feasibility restoration, and occupation-measure projection demonstrate its robustness in highly degenerate settings.
Operator-splitting methods such as the primal-dual hybrid gradient method (PDHG) and the alternating direction method of multipliers (ADMM) often exhibit linear convergence on conic programs, although general theory guarantees only sublinear rates. We identify two geometric conditions -- strict complementarity and quadratic facial violation -- that explain this local behavior: under these conditions, PDHG and ADMM converge linearly to an optimal solution when initialized sufficiently close to the converging strictly complementary solution. We establish this result through a unified and verifiable primal-dual error-bound framework. First, we show that strict complementarity, together with a quadratic facial-violation property of the associated complementary faces, implies uniform quadratic growth of both the primal and dual augmented Lagrangians near a strictly complementary solution. Second, we prove the local equivalence of three regularity conditions: uniform quadratic growth of the augmented Lagrangians, quadratic growth of a localized smoothed primal-dual gap, and metric subregularity of the saddle-point mapping. This equivalence clarifies the relationship among previously proposed conditions for local linear convergence. Third, using a unified formulation, we give a concise analysis showing that these equivalent conditions yield local linear convergence of PDHG and ADMM. We verify the quadratic facial-violation property for standard polyhedral and symmetric cones, as well as relevant faces of exponential and power cones, and show that it is preserved under Cartesian products. We also obtain an improved local rate using a restarted Halpern scheme. Finally, we extend the framework to convex composite optimization through a quadratic subdifferential-violation condition, which generalizes the quadratic facial-violation.
The main contribution is a finite-time mechanism for converting stationarity of the truncated minimax problem into a KKT certificate for the original constrained problem and establishing explicit convergence rates for the proposed method in terms of the KKT residual.
We develop a primal--dual interior-point method for nonsymmetric conic optimization based on a conjugate-free scaling matrix. The scaling is obtained from a single-secant BFGS update of the primal barrier Hessian. In contrast to multi-secant BFGS scalings, it does not require conjugate-barrier derivatives. This feature is important for high-dimensional nonsymmetric cones, where conjugate-barrier derivatives may be unavailable in closed form or expensive to compute. We embed the conjugate-free scaling in a homogeneous self-dual predictor--corrector framework. Using a split central-path neighborhood that separately controls the conic variables and the scalar homogeneous variables, we prove that the scaling matrix remains uniformly comparable to the primal barrier Hessian. This comparison bound is used to prove neighborhood preservation and to show that the complementarity measure and the linear residual decrease at a uniform rate. Consequently, the method attains an iteration bound of $\mathcal{O}(\sqrt{\nu}\log(1/\varepsilon))$, improving the $\mathcal{O}(\nu\log(1/\varepsilon))$ bound of Badenbroek and Dahl [Optim. Methods Softw., 37 (2022), pp. 1027--1064] and matching the best-known complexity order for interior-point methods. Numerical experiments on instances involving the operator perspective epigraph cone and the quantum relative entropy cone show that the method is competitive with QICS, a specialized solver for conic models arising in quantum information.
Primal-dual first-order methods are widely used for large-scale semidefinite programming (SDP), but their ability to compute highly accurate solutions is not well explained by global convergence theory alone. We study the local convergence of the primal-dual hybrid gradient (PDHG) method applied to a standard primal--dual SDP pair. We show that PDHG converges eventually (R-)linearly whenever the limiting KKT point satisfies either strict complementarity or primal--dual nondegeneracy. The proof views PDHG as a preconditioned proximal point method for the KKT inclusion and combines its descent inequality with a local error bound. Under strict complementarity, the error bound follows from the local spectral geometry of the positive semidefinite cone; under primal-dual nondegeneracy, it follows from strong regularity of the KKT mapping. We also give a simple SDP instance where both regularity conditions fail and PDHG can converge only sublinearly. This contrasts with linear programming, where PDHG admits a local linear convergence regime even for degenerate instances. Numerical experiments support the theory and identify difficult SDP instances where PDHG struggles to reach high accuracy.
This work revisits a variant of the Polyak step-size based on Bregman projections due to Kiwiel (1997), and shows that mirror Polyak enjoys guarantees similar to its Euclidean counterpart, automatically adapting to relative notions of smoothness, Lipschitz continuity, or strong convexity.
Frederik Kunstner, Ryan D'Orazio, V. S. Portella et al.· 0 citations
This paper proposes a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable and provides the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems.