arXiv subjects

Lijun Ding

Publications and source records attributed to Lijun Ding.

At least 19 recordsLinked to original sources

arXiv2026

A Triangular Log-Det PDHG for Semidefinite Programming: Complementarity Selection and Global Linear Convergence

Semidefinite programming (SDP) is an important class of convex optimization problems with wide applications in practice. In this paper, we study a solution-selection question for first-order methods for solving SDPs: Can a first-order method provably select a strictly complementary solution when one exists, and a maximally complementary solution otherwise? We propose Triangular Log-Det PDHG (TriLog-PDHG), a primal-dual hybrid gradient method equipped with a triangular log-determinant Bregman kernel. We show that TriLog-PDHG converges to a KKT point for any fixed positive shift and to a maximally complementary solution for every sufficiently small positive shift. When a strictly complementary solution exists, the unshifted method provably selects a strictly complementary KKT point and enjoys a global linear convergence rate. Under the same condition, we further develop an averaged and adaptively restarted variant that accelerates convergence. Numerical experiments on classical examples and SDPLIB instances illustrate the complementarity-selection and convergence properties predicted by the theory.

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arXiv2026

On the Local Linear Convergence of Operator Splitting Methods for Conic Programming

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.

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arXiv2026

Low-Rank Tensor Estimation from Nonlinear Observations: A Unified Framework

We consider the estimation of a $d_1\times d_2\times d_3$ tensor $X^\star$ of Tucker rank $(r_1,r_2,r_3)$ from the nonlinear observations $\{y_i=f_i(\langle A_i,X^\star\rangle)\}_{i=1}^n$. We develop a unified approach that first constructs a gradient map from the data and then establishes the tensor restricted approximate invertibility condition (T-RAIC), a condition that quantifies how well the gradient map aligns with the ideal descent step under a low-rank tensor dual norm. We show that T-RAIC yields local linear convergence guarantees for a Riemannian gradient descent (RGD) algorithm, which may incorporate a normalization step if $\|X^\star\|_{\rm F}$ is known a priori. Under $O(r_1r_2r_3+\sum_{1\le i\le 3}r_id_i)$ Gaussian measurements, we establish T-RAICs for single-index models, logistic regression, phase retrieval, ReLU regression, and one-bit compressed sensing. The RAICs imply that RGD locally converges to $X^\star$ exactly in phase retrieval and ReLU regression, and up to near-optimal estimation errors in the remaining models. We further show that, in all these models except for phase retrieval, a simple spectral initialization yields the desired initialization from $O(d^{3/2})$ measurements under $d_1=d_2=d_3=d$ (ignoring dependence on the Tucker rank and condition number of $X^\star$). This is also the best known sample complexity for polynomial-time and end-to-end algorithms in tensor linear regression and tensor completion. Numerical simulations are provided to corroborate our theoretical findings.

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arXiv2026

On the Complexity of BFGS Method for Smooth Convex Optimization

We study the BFGS method with an Armijo-Wolfe line search for minimizing convex functions with Lipschitz-continuous gradients, without assuming strong convexity. We establish a global iteration complexity bound of $\mathcal{O}(k^{-1/2})$ for the smallest gradient norm among the first $k$ iterates. Moreover, when the initial sublevel set is bounded, we show that the function value gap converges at a rate of $\mathcal{O}(k^{-1})$. Our analysis leverages the classical trace-log-determinant potential function and reveals that a key inequality underlying this potential function remains valid without strong convexity.

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arXiv2026

A Scalable Bundle Method for Exact Reformulation of SDP in Three-Phase Power Flow Feasibility

Power flow feasibility assessment is computationally challenging for unbalanced three-phase distribution networks. This paper develops a vectorized semidefinite program (SDP) based on the bus injection model (BIM) and reformulates its dual as an exact-penalty problem, enabling us to develop a scalable three-cut proximal bundle method for feasibility assessment. The proposed bundle method is numerically over 400 times faster than MOSEK with less than 1/2000 of its memory; on the decomposed BIM-SDP, approximately 2 times faster with 75% less memory.

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arXiv2026

TailedTS: Benchmark Dataset for Heavy-Tailed Time Series Prediction and Periodicity Quantification

We present TailedTS, a large-scale benchmark dataset derived from Wikipedia hourly page view observations throughout 2024, specifically designed to test time series forecasting models under heavy-tailed, zero-inflated, and non-Gaussian conditions. The dataset comprises approximately 24.69 billion data points spanning roughly 3 million unique Wikipedia pages per month, stored in high-efficiency Apache Parquet format. Wikipedia traffic follows a pronounced power-law distribution where roughly 5% of pages account for over 70% of total page views, creating a natural and rigorous testbed for model robustness against extreme volatility that are absent from or underrepresented in existing benchmarks such as M4, M5, and UCI electricity datasets. TailedTS enables several research tasks. First, we introduce a periodicity quantification framework based on sparse autoregression with sparsity and non-negativity constraints, revealing that frequently-viewed pages exhibit significantly weaker periodic structure than their less-viewed counterparts, showing direct implications for server allocation and traffic forecasting on large digital platforms. Second, we provide standardized prediction benchmarks evaluated under a suite of non-Gaussian loss functions, including $\ell_1$-norm, Huber, quantile, and $\ell_p$-norm losses, demonstrating that standard Gaussian-based estimators degrade substantially on high-volume page categories, while robust alternatives provide consistent gains across all traffic scales. TailedTS is publicly available at https://doi.org/10.5281/zenodo.17070469.

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arXiv2026

An Overview and Comparison of Spectral Bundle Methods for Primal and Dual Semidefinite Programs

The spectral bundle method developed by Helmberg and Rendl is well-established for solving large-scale semidefinite programs (SDPs) in the dual form, especially when the SDPs admit $\textit{low-rank primal solutions}$. Under mild regularity conditions, a recent result by Ding and Grimmer has established fast linear convergence rates when the bundle method captures $\textit{the rank of primal solutions}$. In this paper, we present an overview and comparison of spectral bundle methods for solving both $\textit{primal}$ and $\textit{dual}$ SDPs. In particular, we introduce a new family of spectral bundle methods for solving SDPs in the $\textit{primal}$ form. The algorithm developments are parallel to those by Helmberg and Rendl, mirroring the elegant duality between primal and dual SDPs. The new family of spectral bundle methods also achieves linear convergence rates for primal feasibility, dual feasibility, and duality gap when the algorithm captures $\textit{the rank of the dual solutions}$. Therefore, the original spectral bundle method by Helmberg and Rendl is well-suited for SDPs with $\textit{low-rank primal solutions}$, while on the other hand, our new spectral bundle method works well for SDPs with $\textit{low-rank dual solutions}$. These theoretical findings are supported by a range of large-scale numerical experiments. Finally, we demonstrate that our new spectral bundle method achieves state-of-the-art efficiency and scalability for solving polynomial optimization compared to a set of baseline solvers $\textsf{SDPT3}$, $\textsf{MOSEK}$, $\textsf{CDCS}$, and $\textsf{SDPNAL+}$.

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arXiv2026

Error bounds, PL condition, and quadratic growth for weakly convex functions, and linear convergences of proximal point methods

Many practical optimization problems lack strong convexity. Fortunately, recent studies have revealed that first-order algorithms also enjoy linear convergences under various weaker regularity conditions. While the relationship among different conditions for convex and smooth functions is well-understood, it is not the case for the nonsmooth setting. In this paper, we go beyond convexity and smoothness, and clarify the connections among common regularity conditions in the class of weakly convex functions, including $\textit{strong convexity}$, $\textit{restricted secant inequality}$, $\textit{subdifferential error bound}$, $\textit{Polyak-Łojasiewicz inequality}$, and $\textit{quadratic growth}$. In addition, using these regularity conditions, we present a simple and modular proof for the linear convergence of the proximal point method (PPM) for convex and weakly convex optimization problems. The linear convergence also holds when the subproblems of PPM are solved inexactly with a proper control of inexactness.

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arXiv2025

A Validation Approach to Over-parameterized Matrix and Image Recovery

This paper studies the problem of recovering a low-rank matrix from several noisy random linear measurements. We consider the setting where the rank of the ground-truth matrix is unknown a priori and use an objective function built from a rank-overspecified factored representation of the matrix variable, where the global optimal solutions overfit and do not correspond to the underlying ground truth. We then solve the associated nonconvex problem using gradient descent with small random initialization. We show that as long as the measurement operators satisfy the restricted isometry property (RIP) with its rank parameter scaling with the rank of the ground-truth matrix rather than scaling with the overspecified matrix rank, gradient descent iterations are on a particular trajectory towards the ground-truth matrix and achieve nearly information-theoretically optimal recovery when it is stopped appropriately. We then propose an efficient stopping strategy based on the common hold-out method and show that it detects a nearly optimal estimator provably. Moreover, experiments show that the proposed validation approach can also be efficiently used for image restoration with deep image prior, which over-parameterizes an image with a deep network.

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arXiv2025

Interpretable Time Series Autoregression for Periodicity Quantification

Time series autoregression (AR) is a classical tool for modeling auto-correlations and periodic structures in real-world systems. We revisit this model from an interpretable machine learning perspective by introducing sparse autoregression (SAR), where $\ell_0$-norm constraints are used to isolate dominant periodicities. We formulate exact mixed-integer optimization (MIO) approaches for both stationary and non-stationary settings and introduce two scalable extensions: a decision variable pruning (DVP) strategy for temporally-varying SAR (TV-SAR), and a two-stage optimization scheme for spatially- and temporally-varying SAR (STV-SAR). These models enable scalable inference on real-world spatiotemporal datasets. We validate our framework on large-scale mobility and climate time series. On NYC ridesharing data, TV-SAR reveals interpretable daily and weekly cycles as well as long-term shifts due to COVID-19. On climate datasets, STV-SAR uncovers the evolving spatial structure of temperature and precipitation seasonality across four decades in North America and detects global sea surface temperature dynamics, including El Niño. Together, our results demonstrate the interpretability, flexibility, and scalability of sparse autoregression for periodicity quantification in complex time series.

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arXiv2025

New Understandings and Computation on Augmented Lagrangian Methods for Low-Rank Semidefinite Programming

Augmented Lagrangian Method (ALM) combined with Burer-Monteiro (BM) factorization, dubbed ALM-BM, offers a powerful approach for solving large-scale low-rank semidefinite programs (SDPs). Despite its empirical success, the theoretical understandings of the resulting non-convex ALM-BM subproblems, particularly concerning their structural properties and efficient subproblem solvability by first-order methods, still remain limited. This work addresses these notable gaps by providing a rigorous theoretical analysis. We demonstrate that, under appropriate regularity of the original SDP, termed as primal simplicity, ALM subproblems inherit crucial properties such as low-rankness and strict complementarity when the dual variable is localized. Furthermore, ALM subproblems are shown to enjoy a quadratic growth condition, building on which we prove that the non-convex ALM-BM subproblems can be solved to global optimality by gradient descent, achieving linear convergence under conditions of local initialization and dual variable proximity. Through illustrative examples, we further establish the necessity of these local assumptions, revealing them as inherent characteristics of the problem structure. Motivated by these theoretical insights, we propose ALORA, a rank-adaptive augmented Lagrangian method that builds upon the ALM-BM framework, which dynamically adjusts the rank using spectral information and explores negative curvature directions to navigate the nonconvex landscape. Exploiting modern GPU computing architectures, ALORA exhibits strong numerical performance, solving SDPs with tens of millions of dimensions in hundreds of seconds.

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arXiv2025

On Squared-Variable Formulations for Nonlinear Semidefinite programming

In optimization problems involving smooth functions and real and matrix variables, that contain matrix semidefiniteness constraints, consider the following change of variables: Replace the positive semidefinite matrix $X \in \mathbb{S}^d$, where $\mathbb{S}^d$ is the set of symmetric matrices in $\mathbb{R}^{d\times d}$, by a matrix product $FF^\top$, where $F \in \mathbb{R}^{d \times d}$ or $F \in \mathbb{S}^d$. The formulation obtained in this way is termed ``squared variable," by analogy with a similar idea that has been proposed for real (scalar) variables. It is well known that points satisfying first-order conditions for the squared-variable reformulation do not necessarily yield first-order points for the original problem. There are closer correspondences between second-order points for the squared-variable reformulation and the original formulation. These are explored in this paper, along with correspondences between local minimizers of the two formulations.

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arXiv2024

On Squared-Variable Formulations

We revisit a formulation technique for inequality constrained optimization problems that has been known for decades: the substitution of squared variables for nonnegative variables. Using this technique, inequality constraints are converted to equality constraints via the introduction of a squared-slack variable. Such formulations have the superficial advantage that inequality constraints can be dispensed with altogether. But there are clear disadvantages, not least being that first-order optimal points for the squared-variable reformulation may not correspond to first-order optimal points for the original problem, because the Lagrange multipliers may have the wrong sign. Extending previous results, this paper shows that points satisfying approximate second-order optimality conditions for the squared-variable reformulation also, under certain conditions, satisfy approximate second-order optimality conditions for the original formulation, and vice versa. Such results allow us to adapt complexity analysis of methods for equality constrained optimization to account for inequality constraints. On the algorithmic side, we examine squared-variable formulations for several interesting problem classes, including bound-constrained quadratic programming, linear programming, and nonnegative matrix factorization. We show that algorithms built on these formulations are surprisingly competitive with standard methods. For linear programming, we examine the relationship between the squared-variable approach and primal-dual interior-point methods.

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arXiv2024

Inexact Augmented Lagrangian Methods for Conic Programs: Quadratic Growth and Linear Convergence

Augmented Lagrangian Methods (ALMs) are widely employed in solving constrained optimizations, and some efficient solvers are developed based on this framework. Under the quadratic growth assumption, it is known that the dual iterates and the Karush-Kuhn-Tucker (KKT) residuals of ALMs applied to semidefinite programs (SDPs) converge linearly. In contrast, the convergence rate of the primal iterates has remained elusive. In this paper, we resolve this challenge by establishing new $\textit{quadratic growth}$ and $\textit{error bound}$ properties for primal and dual SDPs under the strict complementarity condition. Our main results reveal that both primal and dual iterates of the ALMs converge linearly contingent solely upon the assumption of strict complementarity and a bounded solution set. This finding provides a positive answer to an open question regarding the asymptotically linear convergence of the primal iterates of ALMs applied to semidefinite optimization.

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arXiv2024

Sharpness and well-conditioning of nonsmooth convex formulations in statistical signal recovery

We study a sample complexity vs. conditioning tradeoff in modern signal recovery problems (including sparse recovery, low-rank matrix sensing, covariance estimation, and abstract phase retrieval), where convex optimization problems are built from sampled observations. We begin by introducing a set of condition numbers related to sharpness in $\ell_p$ or Schatten-$p$ norms ($p\in[1,2]$) of a nonsmooth formulation for these problems. Then, we show that these condition numbers become dimension independent constants in each of the example signal recovery problems once the sample size exceeds some constant multiple of the recovery threshold. Structurally, this result ensures that the inaccuracy in the recovered signal due to both observation noise and optimization error is well-controlled. Algorithmically, such a result ensures that a new restarted mirror descent method achieves nearly-dimension-independent linear convergence to the signal. This new first-order method is general and applies to any sharp convex function in an $\ell_p$ or Schatten-$p$ norm ($p\in[1,2]$).

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arXiv2023

How Over-Parameterization Slows Down Gradient Descent in Matrix Sensing: The Curses of Symmetry and Initialization

This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where $M^* \in \mathbb{R}^{n \times n}$ is a positive semi-definite unknown matrix of rank $r \ll n$, and one uses a symmetric parameterization $XX^\top$ to learn $M^*$. Here $X \in \mathbb{R}^{n \times k}$ with $k > r$ is the factor matrix. We give a novel $Ω(1/T^2)$ lower bound of randomly initialized GD for the over-parameterized case ($k >r$) where $T$ is the number of iterations. This is in stark contrast to the exact-parameterization scenario ($k=r$) where the convergence rate is $\exp (-Ω(T))$. Next, we study asymmetric setting where $M^* \in \mathbb{R}^{n_1 \times n_2}$ is the unknown matrix of rank $r \ll \min\{n_1,n_2\}$, and one uses an asymmetric parameterization $FG^\top$ to learn $M^*$ where $F \in \mathbb{R}^{n_1 \times k}$ and $G \in \mathbb{R}^{n_2 \times k}$. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case ($k=r$) with an $\exp (-Ω(T))$ rate. Furthermore, we give the first global exact convergence result for the over-parameterization case ($k>r$) with an $\exp(-Ω(α^2 T))$ rate where $α$ is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from $Ω(1/T^2)$ to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of $α$, recovering the rate in the exact-parameterization case.

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arXiv2023

Euclidean-Norm-Induced Schatten-p Quasi-Norm Regularization for Low-Rank Tensor Completion and Tensor Robust Principal Component Analysis

The nuclear norm and Schatten-$p$ quasi-norm are popular rank proxies in low-rank matrix recovery. However, computing the nuclear norm or Schatten-$p$ quasi-norm of a tensor is hard in both theory and practice, hindering their application to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). In this paper, we propose a new class of tensor rank regularizers based on the Euclidean norms of the CP component vectors of a tensor and show that these regularizers are monotonic transformations of tensor Schatten-$p$ quasi-norm. This connection enables us to minimize the Schatten-$p$ quasi-norm in LRTC and TRPCA implicitly via the component vectors. The method scales to big tensors and provides an arbitrarily sharper rank proxy for low-rank tensor recovery compared to the nuclear norm. On the other hand, we study the generalization abilities of LRTC with the Schatten-$p$ quasi-norm regularizer and LRTC with the proposed regularizers. The theorems show that a relatively sharper regularizer leads to a tighter error bound, which is consistent with our numerical results. Particularly, we prove that for LRTC with Schatten-$p$ quasi-norm regularizer on $d$-order tensors, $p=1/d$ is always better than any $p>1/d$ in terms of the generalization ability. We also provide a recovery error bound to verify the usefulness of small $p$ in the Schatten-$p$ quasi-norm for TRPCA. Numerical results on synthetic data and real data demonstrate the effectiveness of the regularization methods and theorems.

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arXiv2023

Provably Convergent Policy Optimization via Metric-aware Trust Region Methods

Trust-region methods based on Kullback-Leibler divergence are pervasively used to stabilize policy optimization in reinforcement learning. In this paper, we exploit more flexible metrics and examine two natural extensions of policy optimization with Wasserstein and Sinkhorn trust regions, namely Wasserstein policy optimization (WPO) and Sinkhorn policy optimization (SPO). Instead of restricting the policy to a parametric distribution class, we directly optimize the policy distribution and derive their closed-form policy updates based on the Lagrangian duality. Theoretically, we show that WPO guarantees a monotonic performance improvement, and SPO provably converges to WPO as the entropic regularizer diminishes. Moreover, we prove that with a decaying Lagrangian multiplier to the trust region constraint, both methods converge to global optimality. Experiments across tabular domains, robotic locomotion, and continuous control tasks further demonstrate the performance improvement of both approaches, more robustness of WPO to sample insufficiency, and faster convergence of SPO, over state-of-art policy gradient methods.

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