arXiv subjects

Kangkang Deng

Publications and source records attributed to Kangkang Deng.

At least 19 recordsLinked to original sources

arXiv2026

A Fixed-Penalty Linearized Augmented Lagrangian Method with Classical Multiplier Updates

Augmented Lagrangian methods are effective for nonlinear equality-constrained optimization, but solving their nonlinear primal subproblems can be expensive. For smooth nonconvex problems with deterministic or stochastic objectives, we propose a nonlinear-residual linearized augmented Lagrangian method (NR-LALM) that replaces this subproblem by a regularized Gauss-Newton-type step while retaining the classical multiplier update based on the nonlinear constraint residual. The resulting step is computed from one symmetric positive-definite linear system, but the mismatch between the linearized primal model and the nonlinear-residual update produces a quadratic constraint-linearization error in the multiplier identity. We show that this error can be controlled under local regularity; multiplier boundedness and trajectory localization are derived rather than assumed. With fixed, accuracy-independent parameters, deterministic NR-LALM finds an $\varepsilon$-approximate Karush-Kuhn-Tucker (KKT) pair in $O(\varepsilon^{-2})$ iterations and first-order oracle evaluations. For stochastic objectives, a projected stochastic path-integrated differential estimator with safeguarded restarts requires, in expectation, $O(\varepsilon^{-3})$ stochastic-gradient evaluations and $O(\varepsilon^{-2})$ constraint and Jacobian evaluations. Compactness and a Kurdyka-Lojasiewicz condition further yield finite-length convergence of the deterministic primal-dual sequence. An optional minimum-norm second-order correction reduces the constraint-linearization error from second to fourth order without changing the complexity orders. All theoretical results are formalized in Lean 4. Numerical experiments confirm the predicted error orders and show favorable performance on high-dimensional deterministic and stochastic problems.

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arXiv2025

A Single-loop Stochastic Riemannian ADMM for Nonsmooth Optimization

We study a class of nonsmooth stochastic optimization problems on Riemannian manifolds. In this work, we propose MARS-ADMM, the first stochastic Riemannian alternating direction method of multipliers with provable near-optimal complexity guarantees. Our algorithm incorporates a momentum-based variance-reduced gradient estimator applied exclusively to the smooth component of the objective, together with carefully designed penalty parameter and dual stepsize updates. Unlike existing approaches that rely on computationally expensive double-loop frameworks, MARS-ADMM operates in a single-loop fashion and requires only a constant number of stochastic gradient evaluations per iteration. Under mild assumptions, we establish that MARS-ADMM achieves an iteration complexity of $\tilde{\mathcal{O}}(\varepsilon^{-3})$, which improves upon the previously best-known bound of $\mathcal{O}(\varepsilon^{-3.5})$ for stochastic Riemannian operator-splitting methods. As a result, our analysis closes the theoretical complexity gap between stochastic Riemannian operator-splitting algorithms and stochastic methods for nonsmooth optimization with nonlinear constraints. Notably, the obtained complexity also matches the best-known bounds in deterministic nonsmooth Riemannian optimization, demonstrating that deterministic-level accuracy can be achieved using only constant-size stochastic samples.

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arXiv2025

Adaptive Riemannian ADMM for Nonsmooth Optimization: Optimal Complexity without Smoothing

We study the problem of minimizing the sum of a smooth function and a nonsmooth convex regularizer over a compact Riemannian submanifold embedded in Euclidean space. By introducing an auxiliary splitting variable, we propose an adaptive Riemannian alternating direction method of multipliers (ARADMM), which, for the first time, achieves convergence without requiring smoothing of the nonsmooth term. Our approach involves only one Riemannian gradient evaluation and one proximal update per iteration. Through careful and adaptive coordination of the stepsizes and penalty parameters, we establish an optimal iteration complexity of order $\mathcal{O}(ε^{-3})$ for finding an $ε$-approximate KKT point, matching the complexity of existing smoothing technique-based Riemannian ADMM methods. Extensive numerical experiments on sparse PCA and robust subspace recovery demonstrate that our ARADMM consistently outperforms state-of-the-art Riemannian ADMM variants in convergence speed and solution quality.

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arXiv2025

The Augmented Lagrangian Methods: Overview and Recent Advances

Large-scale constrained optimization is pivotal in modern scientific, engineering, and industrial computation, often involving complex systems with numerous variables and constraints. This paper provides a unified and comprehensive perspective on constructing augmented Lagrangian functions (based on Hestenes-Powell-Rockafellar augmented Lagrangian) for various optimization problems, including nonlinear programming and convex and nonconvex composite programming. We present the augmented Lagrangian method (ALM), covering its theoretical foundations in both convex and nonconvex cases, and discuss several successful examples and applications. Recent advancements have extended ALM's capabilities to handle nonconvex constraints and ensure global convergence to first and second-order stationary points. For nonsmooth convex problems, ALM utilizes proximal operations, preserving desirable properties such as locally linear convergence rates. Furthermore, recent progress has refined the complexity analysis for ALM and tackled challenging integer programming instances. This review aims to offer a thorough understanding of ALM's benefits and limitations, exploring different ALM variants designed to enhance convergence and computational performance. We also illustrate effective algorithms for ALM subproblems across different types of optimization problems and highlight practical implementations in several fields.

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arXiv2025

Stochastic ADMM with batch size adaptation for nonconvex nonsmooth optimization

Stochastic alternating direction method of multipliers (SADMM) is a popular method for solving nonconvex nonsmooth optimization in various applications. However, it typically requires an empirical selection of the static batch size for gradient estimation, resulting in a challenging trade-off between variance reduction and computational cost. This paper proposes adaptive batch size SADMM, a practical method that dynamically adjusts the batch size based on accumulated differences along the optimization path. We develop a simple convergence analysis to handle the dependence of batch size adaptation that matches the best-known complexity with flexible parameter choices. We further extend this adaptive scheme to reduce the overall complexity of the popular variance-reduced methods, SVRG-ADMM and SPIDER-ADMM. Numerical results validate the effectiveness of our proposed methods.

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arXiv2025

Single-loop $\mathcal{O}(ε^{-3})$ stochastic smoothing algorithms for nonsmooth Riemannian optimization

In this paper, we develop two Riemannian stochastic smoothing algorithms for nonsmooth optimization problems on Riemannian manifolds, addressing distinct forms of the nonsmooth term $ h $. Both methods combine dynamic smoothing with a momentum-based variance reduction scheme in a fully online manner. When $ h $ is Lipschitz continuous, we propose an stochastic algorithm under adaptive parameter that achieves the optimal iteration complexity of $ \mathcal{O}(ε^{-3}) $, improving upon the best-known rates for exist algorithms. When $ h $ is the indicator function of a convex set, we design a new algorithm using truncated momentum, and under a mild error bound condition with parameter $ θ\geq 1 $, we establish a complexity of $ \tilde{\mathcal{O}}(ε^{-\max\{θ+2, 2θ\}}) $, in line with the best-known results in the Euclidean setting. Both algorithms feature a single-loop design with low per-iteration cost and require only $ \mathcal{O}(1) $ samples per iteration, ensuring that sample and iteration complexities coincide. Our framework encompasses a broad class of problems and recovers or matches optimal complexity guarantees in several important settings, including smooth stochastic Riemannian optimization, composite problems in Euclidean space, and constrained optimization via indicator functions.

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arXiv2025

Anderson acceleration of derivative-free projection methods for constrained monotone nonlinear equations

The derivative-free projection method (DFPM) is an efficient algorithm for solving monotone nonlinear equations. As problems grow larger, there is a strong demand for speeding up the convergence of DFPM. This paper considers the application of Anderson acceleration (AA) to DFPM for constrained monotone nonlinear equations. By employing a nonstationary relaxation parameter and interleaving with slight modifications in each iteration, a globally convergent variant of AA for DFPM named as AA-DFPM is proposed. Further, the linear convergence rate is proved under some mild assumptions. Experiments on both mathematical examples and a real-world application show encouraging results of AA-DFPM and confirm the suitability of AA for accelerating DFPM in solving optimization problems.

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arXiv2025

An efficient primal dual semismooth Newton method for semidefinite programming

In this paper, we present an efficient semismooth Newton method, named SSNCP, for solving a class of semidefinite programming problems. Our approach is rooted in an equivalent semismooth system derived from the saddle point problem induced by the augmented Lagrangian duality. An additional correction step is incorporated after the semismooth Newton step to ensure that the iterates eventually reside on a manifold where the semismooth system is locally smooth. Global convergence is achieved by carefully designing inexact criteria and leveraging the $α$-averaged property to analyze the error. The correction steps address challenges related to the lack of smoothness in local convergence analysis. Leveraging the smoothness established by the correction steps and assuming a local error bound condition, we establish the local superlinear convergence rate without requiring the stringent assumptions of nonsingularity or strict complementarity. Furthermore, we prove that SSNCP converges to an $\varepsilon$-stationary point with an iteration complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-3/2})$. Numerical experiments on various datasets, especially the Mittelmann benchmark, demonstrate the high efficiency and robustness of SSNCP compared to state-of-the-art solvers.

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arXiv2025

Stochastic momentum ADMM for nonconvex and nonsmooth optimization with application to PnP algorithm

This paper proposes SMADMM, a single-loop Stochastic Momentum Alternating Direction Method of Multipliers for solving a class of nonconvex and nonsmooth composite optimization problems. SMADMM achieves the optimal oracle complexity of $\mathcal{O}(ε^{-3/2})$ in the online setting. Unlike previous stochastic ADMM algorithms that require large mini-batches or a double-loop structure, SMADMM uses only $\mathcal{O}(1)$ stochastic gradient evaluations per iteration and avoids costly restarts. To further improve practicality, we incorporate dynamic step sizes and penalty parameters, proving that SMADMM maintains its optimal complexity without the need for large initial batches. We also develop PnP-SMADMM by integrating plug-and-play priors, and establish its theoretical convergence under mild assumptions. Extensive experiments on classification, CT image reconstruction, and phase retrieval tasks demonstrate that our approach outperforms existing stochastic ADMM methods both in accuracy and efficiency, validating our theoretical results.

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arXiv2024

Decentralized projected Riemannian stochastic recursive momentum method for nonconvex optimization

This paper studies decentralized optimization over a compact submanifold within a communication network of $n$ nodes, where each node possesses a smooth non-convex local cost function, and the goal is to jointly minimize the sum of these local costs. We focus particularly on the online setting, where local data is processed in real-time as it streams in, without the need for full data storage. We propose a decentralized projected Riemannian stochastic recursive momentum (DPRSRM) method that employs local hybrid stochastic gradient estimators and uses the network to track the global gradient. DPRSRM achieves an oracle complexity of $\mathcal{O}(\epsilon^{-\frac{3}{2}})$, outperforming existing methods that have at most $\mathcal{O}(\epsilon^{-2})$ complexity. Our method requires only $\mathcal{O}(1)$ gradient evaluations per iteration for each local node and does not require restarting with a large batch gradient. Furthermore, we demonstrate the effectiveness of our proposed methods compared to state-of-the-art ones through numerical experiments on principal component analysis problems and low-rank matrix completion.

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arXiv2024

Inexact Riemannian Gradient Descent Method for Nonconvex Optimization

Gradient descent methods are fundamental first-order optimization algorithms in both Euclidean spaces and Riemannian manifolds. However, the exact gradient is not readily available in many scenarios. This paper proposes a novel inexact Riemannian gradient descent algorithm for nonconvex problems, accompanied by a convergence guarantee. In particular, we establish two inexact gradient conditions on Riemannian manifolds for the first time, enabling precise gradient approximations. Our method demonstrates strong convergence results for both gradient sequences and function values. The global convergence with constructive convergence rates for the sequence of iterates is ensured under the Riemannian Kurdyka-Łojasiewicz property. Furthermore, our algorithm encompasses two specific applications: Riemannian sharpness-aware minimization and Riemannian extragradient algorithm, both of which inherit the global convergence properties of the inexact gradient methods. Numerical experiments on low-rank matrix completion and principal component analysis problems validate the efficiency and practical relevance of the proposed approaches.

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arXiv2024

Improving the communication in decentralized manifold optimization through single-step consensus and compression

We are concerned with decentralized optimization over a compact submanifold, where the loss functions of local datasets are defined by their respective local datasets. A key challenge in decentralized optimization is mitigating the communication bottleneck, which primarily involves two strategies: achieving consensus and applying communication compression. Existing projection/retraction-type algorithms rely on multi-step consensus to attain both consensus and optimality. Due to the nonconvex nature of the manifold constraint, it remains an open question whether the requirement for multi-step consensus can be reduced to single-step consensus. We address this question by carefully elaborating on the smoothness structure and the asymptotic 1-Lipschitz continuity associated with the manifold constraint. Furthermore, we integrate these insights with a communication compression strategy to propose a communication-efficient gradient algorithm for decentralized manifold optimization problems, significantly reducing per-iteration communication costs. Additionally, we establish an iteration complexity of $\mathcal{O}(ε^{-1})$ to find an $ε$-stationary point, which matches the complexity in the Euclidean setting. Numerical experiments demonstrate the efficiency of the proposed method in comparison to state-of-the-art approaches.

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arXiv2024

New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring-Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak-Ribi$\grave{e}$re-Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.

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arXiv2024

An Augmented Lagrangian Primal-Dual Semismooth Newton Method for Multi-Block Composite Optimization

In this paper, we develop a novel primal-dual semismooth Newton method for solving linearly constrained multi-block convex composite optimization problems. First, a differentiable augmented Lagrangian (AL) function is constructed by utilizing the Moreau envelopes of the nonsmooth functions. It enables us to derive an equivalent saddle point problem and establish the strong AL duality under the Slater's condition. Consequently, a semismooth system of nonlinear equations is formulated to characterize the optimality of the original problem instead of the inclusion-form KKT conditions. We then develop a semismooth Newton method, called ALPDSN, which uses purely second-order steps and a nonmonotone line search based globalization strategy. Through a connection to the inexact first-order steps when the regularization parameter is sufficiently large, the global convergence of ALPDSN is established. Under the regularity conditions, partial smoothness, the local error bound, and the strict complementarity, we show that both the primal and the dual iteration sequences possess a superlinear convergence rate and provide concrete examples where these regularity conditions are met. Numerical results on the image restoration with two regularization terms and the corrected tensor nuclear norm problem are presented to demonstrate the high efficiency and robustness of our ALPDSN.

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arXiv2024

Oracle complexities of augmented Lagrangian methods for nonsmooth manifold optimization

In this paper, we present two novel manifold inexact augmented Lagrangian methods, \textbf{ManIAL} for deterministic settings and \textbf{StoManIAL} for stochastic settings, solving nonsmooth manifold optimization problems. By using the Riemannian gradient method as a subroutine, we establish an $\mathcal{O}(ε^{-3})$ oracle complexity result of \textbf{ManIAL}, matching the best-known complexity result. Our algorithm relies on the careful selection of penalty parameters and the precise control of termination criteria for subproblems. Moreover, for cases where the smooth term follows an expectation form, our proposed \textbf{StoManIAL} utilizes a Riemannian recursive momentum method as a subroutine, and achieves an oracle complexity of $\tilde{\mathcal{O}}(ε^{-3.5})$, which surpasses the best-known $\mathcal{O}(ε^{-4})$ result. Numerical experiments conducted on sparse principal component analysis and sparse canonical correlation analysis demonstrate that our proposed methods outperform an existing method with the previously best-known complexity result. To the best of our knowledge, these are the first complexity results of the augmented Lagrangian methods for solving nonsmooth manifold optimization problems.

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arXiv2024

A cut-and-project perspective for linearized Bregman iterations

The linearized Bregman iterations (LBreI) and its variants are powerful tools for finding sparse or low-rank solutions to underdetermined linear systems. In this study, we propose a cut-and-project perspective for the linearized Bregman method via a bilevel optimization formulation, along with a new unified algorithmic framework. The new perspective not only encompasses various existing linearized Bregman iteration variants as specific instances, but also allows us to extend the linearized Bregman method to solve more general inverse problems. We provide a completed convergence result of the proposed algorithmic framework, including convergence guarantees to feasible points and optimal solutions, and the sublinear convergence rate. Moreover, we introduce the Bregman distance growth condition to ensure linear convergence. At last, our findings are illustrated via numerical tests.

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arXiv2023

Decentralized Douglas-Rachford splitting methods for smooth optimization over compact submanifolds

We study decentralized smooth optimization problems over compact submanifolds. Recasting it as a composite optimization problem, we propose a decentralized Douglas-Rachford splitting algorithm, DDRS. When the proximal operator of the local loss function does not have a closed-form solution, an inexact version of DDRS, iDDRS, is also presented. Both algorithms rely on an ingenious integration of the nonconvex Douglas-Rachford splitting algorithm with gradient tracking and manifold optimization. We show that our DDRS and iDDRS achieve the best-known convergence rate of $\mathcal{O}(1/K)$. The main challenge in the proof is how to handle the nonconvexity of the manifold constraint. To address this issue, we utilize the concept of proximal smoothness for compact submanifolds. This ensures that the projection onto the submanifold exhibits convexity-like properties, which allows us to control the consensus error across agents. Numerical experiments on the principal component analysis are conducted to demonstrate the effectiveness of our decentralized DRS compared with the state-of-the-art ones.

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arXiv2023

Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

In this paper, we introduce the notion of generalized $ε$-stationarity for a class of nonconvex and nonsmooth composite minimization problems on compact Riemannian submanifold embedded in Euclidean space. To find a generalized $ε$-stationarity point, we develop a family of Riemannian gradient-type methods based on the Moreau envelope technique with a decreasing sequence of smoothing parameters, namely Riemannian smoothing gradient and Riemannian smoothing stochastic gradient methods. We prove that the Riemannian smoothing gradient method has the iteration complexity of $\mathcal{O}(ε^{-3})$ for driving a generalized $ε$-stationary point. To our knowledge, this is the best-known iteration complexity result for the nonconvex and nonsmooth composite problem on manifolds. For the Riemannian smoothing stochastic gradient method, one can achieve the iteration complexity of $\mathcal{O}(ε^{-5})$ for driving a generalized $ε$-stationary point. Numerical experiments are conducted to validate the superiority of our algorithms.

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