arXiv subjects

Huikang Liu

Publications and source records attributed to Huikang Liu.

At least 19 recordsLinked to original sources

arXiv2026

GPU-Accelerated Conic Quadratic Programming with Local Linear Convergence under Strict Complementarity

We present PDHCG-CQP, a GPU-accelerated first-order solver for large-scale conic convex quadratic programming. PDHCG-CQP supports affine constraints and Cartesian products of nonnegative, second-order, rotated second-order, exponential, and three-dimensional power cones. At its core is a restarted averaged primal-dual hybrid gradient (PDHG) method, whose primal update is computed inexactly by solving a conic quadratic proximal subproblem with projected gradient iterations. We establish local linear convergence of the restarted averaged scheme with both exact and inexact primal proximal evaluations under a uniform local quadratic-growth condition on the smoothed primal-dual gap. We further show that this condition holds under strict complementarity by exploiting a rotated second-order-cone lifting together with local primal and dual regularity conditions. Our C/CUDA implementation combines matrix-free linear algebra, batched cone projections, adaptive inner solves, reflected-Halpern acceleration, and fully device-resident KKT residual computations. It also supports multi-GPU execution through a two-dimensional partitioning of the problem data. Extensive experiments on standard and large-scale quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP), and quasilinear Fisher equilibrium benchmarks demonstrate that PDHCG-CQP achieves state-of-the-art robustness among first-order solvers while scaling efficiently to 8 GPUs and instances with up to $4.4\times10^8$ stored primal coordinates. PDHCG-CQP is open source and available at https://github.com/Lhongpei/PDHCG.

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arXiv2026

A Curvature-Aware Rank-Adaptive Distributed Augmented-Lagrangian Solver for Large-Scale SDPs

We present CARDAL (Curvature-Aware Rank-Adaptive Distributed Augmented Lagrangian), a distributed multi-GPU solver for large-scale semidefinite programs (SDPs) based on a rank-adaptive Burer-Monteiro factorization and an augmented Lagrangian method. At fixed ranks, a matrix-free L-BFGS method with negative-curvature corrections targets an approximate Euclidean second-order stationary point of the factored augmented Lagrangian. A reverse multiplier shift turns a negative dual-slack direction into exact negative curvature after rank expansion, and a small joint rank-lift problem selects a batched low-rank correction. A verified slack lower bound provides an a posteriori approximate KKT certificate. Our analysis establishes generic global-optimality guarantees for heterogeneous products of PSD cones at per-block ranks near the Barvinok-Pataki scale, together with a finite-accuracy counterpart under blockwise cost smoothing. For scalable execution, CARDAL distributes constraint rows, factor columns, and PSD blocks over a Constraint x Rank x Cone device mesh. The primal residual, gradient, Hessian-vector products, and slack matrix-vector products are evaluated using device-local operations and axis-wise collectives. On the Mittelmann benchmark, CARDAL exhibits stronger robustness than existing low-rank GPU approaches under a uniform accuracy standard. Experiments on large-scale SDP relaxations from robotics, electronic structure, and Max-Cut demonstrate the complementary scaling regimes of the three distribution axes, with observed wall-clock speedups of up to 4x on four H100 GPUs.

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arXiv2026

A Concentration Inequality for the Covariance Matrix of an Arbitrary Subset of Random Vectors

Concentration inequalities for sample covariance matrices are fundamental tools in high-dimensional probability. Classical results typically assume that the selected random vectors are independent of the selection rule. In this paper, we study spectral concentration for sample covariance matrices formed from arbitrary, possibly data-dependent subsets of i.i.d. random vectors. Such data-dependent selection destroys the usual independence structure and makes standard covariance concentration bounds inapplicable. For i.i.d. Gaussian random vectors, we prove high-probability lower and upper bounds for the minimal and maximal eigenvalues of such selected covariance matrices. Compared with a direct union-bound argument, our results provide substantially sharper guarantees and allow much smaller subset proportions. We further discuss extensions from Gaussian to sub-Gaussian random vectors, and beyond independence to weakly dependent observations, with geometrically strong-mixing Gaussian sequences serving as a representative example of the latter. Finally, we apply the developed concentration inequalities to the K-subspace clustering problem under a low-rank Gaussian mixture model, where the optimal clusters are inherently data-dependent. Our results yield recovery guarantees showing that the clustering error of global minimizers decays polynomially with the signal-to-noise ratio.

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arXiv2026

Mind the Gap: Mixtures of Gaussians in Approximate Differential Privacy

We design a class of additive noise mechanisms that satisfy $(\varepsilon, \delta)$-differential privacy (DP) for scalar, real-valued query functions with known sensitivities, with a particular focus on moderate and low-privacy regimes. These mechanisms, which we call \textit{mixture mechanisms}, are constructed by mixing multiple Gaussian distributions that share the same variance but differ in their means and mixture weights. The resulting distributions can be interpreted as convex combinations of a zero-mean Gaussian (as used in the analytic Gaussian mechanism) and additional Gaussians whose means depend on the sensitivity of the query function. We derive tight conditions on the variances required for $(\varepsilon, \delta)$-DP and provide efficient algorithms to compute them. Compared to the analytic Gaussian mechanism, our mechanisms yield substantially lower expected noise amplitudes ($l_1$-loss) and variances ($l_2$-loss for zero-mean distributions). In the low-privacy regime that motivates our design, our mechanisms approach optimality, mitigating nearly all of the optimality gap of the analytic Gaussian mechanism.

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arXiv2026

On the convergence of doubly stochastic Primal-Dual Hybrid Gradient Method

We study a block-structured class of convex-concave saddle-point problems in which both the primal and dual variables admit natural separable decompositions. Motivated by large-scale applications where a full update on either side can be computationally expensive, we propose a doubly stochastic primal--dual hybrid gradient method (DSPDHG) that performs randomized block updates on both primal and dual variables.The method extends classical PDHG and stochastic PDHG (SPDHG) schemes in a unified manner:it reduces to deterministic PDHG when all blocks are selected and to one-sided stochastic variants when only one side is randomized. For the general convex setting, we establish an $\mathcal{O}(1/K)$ ergodic convergence rate for the expected restricted primal--dual gap under suitable blockwise step-size conditions. We further analyze a restarted variant of DSPDHG under a quadratic growth condition in terms of the smoothed primal-dual gap. Under this regularity assumption, we prove linear convergence of the restarted outer iterates. Numerical evidence is provided to show that restarted DSPDHG with standard step sizes demonstrates competitive practical performance compared with PDHG, SPDHG, and their restarted variants.

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arXiv2026

Anchored Spectral Estimator for Rigid Motion Synchronization

A rigid motion in $\mathbb{R}^d$ consists of a proper rotation and a translation, and it can be represented as a matrix in $\mathbb{R}^{(d+1)\times (d+1)}$. The problem of rigid motion synchronization aims to estimate a collection of rigid motions $G^*_1, \dots, G^*_n$ from noisy observations of their comparisons ${G^*_i}^{-1} G^*_j$. Such problems naturally arise in diverse applications across signal processing, robotics, and computer vision, and have thus attracted intense research attention in recent years. Motivated by geometric considerations, this paper develops a novel spectral approach for rigid motion synchronization, called the anchored spectral estimator (ASE). Theoretically, we establish uniform estimation error bounds for the estimators produced by ASE. Empirically, we show that ASE outperforms the widely used two-stage approach, which first estimates the rotations and then the translations. Further numerical experiments on the multiple point-set registration problem are presented to demonstrate the superiority of ASE over state-of-the-art methods.

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arXiv2026

PDHCG-II: An Enhanced Version of PDHCG for Large-Scale Convex QP

Quadratic programming (QP) is a fundamental optimization model with wide-ranging applications in decision-making and machine learning, yet efficiently solving large-scale instances remains a major computational challenge. Building upon the recently developed PDHCG framework, we propose PDHCG-II, an enhanced first-order solver tailored for large-scale convex QPs. The proposed method explicitly exploits the quadratic structure of the objective and incorporates several key algorithmic innovations, including Halpern-type acceleration and a PID-controlled adaptive update of the primal-dual weight. To further improve practical performance, PDHCG-II introduces a refined adaptive termination criterion for inner subproblems to prevent over-solving, together with an infeasibility detection mechanism for robust handling of ill-posed instances. Extensive numerical experiments demonstrate that PDHCG-II consistently achieves 2.5-5 times speedups over PDHCG on standard QP benchmarks. To facilitate reproducibility and broader adoption, we release a CUDA-C implementation of PDHCG-II as open-source software.

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arXiv2026

D-PDLP: Scaling PDLP to Distributed Multi-GPU Systems

We present a distributed framework of the Primal-Dual Hybrid Gradient (PDHG) algorithm for solving massive-scale linear programming (LP) problems. Although PDHG-based solvers demonstrate strong performance on single-node GPU architectures, their applicability to industrial-scale instances is often limited by single-GPU computational throughput. To overcome these challenges, we propose D-PDLP, the first Distributed PDLP framework, which extends PDHG to a multi-GPU setting via a practical two-dimensional grid partitioning of the constraint matrix. To improve load balance and computational efficiency, we introduce a block-wise random permutation strategy combined with nonzero-aware matrix partitioning. By distributing the intensive computation required in PDHG iterations, the proposed framework harnesses multi-GPU parallelism to achieve substantial speedups with relatively low communication overhead. Extensive experiments on standard LP benchmarks (including MIPLIB and Mittelmann instances) as well as huge-scale real-world datasets show that our distributed implementation, built upon cuPDLPx, achieves strong scalability and high performance while preserving full FP64 numerical accuracy.

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arXiv2025

A Global Geometric Analysis of Maximal Coding Rate Reduction

The maximal coding rate reduction (MCR$^2$) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective deep network architectures. However, it lacks a complete theoretical justification: only the properties of its global optima are known, and its global landscape has not been studied. In this work, we give a complete characterization of the properties of all its local and global optima, as well as other types of critical points. Specifically, we show that each (local or global) maximizer of the MCR$^2$ problem corresponds to a low-dimensional, discriminative, and diverse representation, and furthermore, each critical point of the objective is either a local maximizer or a strict saddle point. Such a favorable landscape makes MCR$^2$ a natural choice of objective for learning diverse and discriminative representations via first-order optimization methods. To validate our theoretical findings, we conduct extensive experiments on both synthetic and real data sets.

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arXiv2025

OptPipe: Memory- and Scheduling-Optimized Pipeline Parallelism for LLM Training

Pipeline parallelism (PP) has become a standard technique for scaling large language model (LLM) training across multiple devices. However, despite recent progress in reducing memory consumption through activation offloading, existing approaches remain largely heuristic and coarse-grained, often overlooking the fine-grained trade-offs between memory, computation, and scheduling latency. In this work, we revisit the pipeline scheduling problem from a principled optimization perspective. We observe that prevailing strategies either rely on static rules or aggressively offload activations without fully leveraging the interaction between memory constraints and scheduling efficiency. To address this, we formulate scheduling as a constrained optimization problem that jointly accounts for memory capacity, activation reuse, and pipeline bubble minimization. Solving this model yields fine-grained schedules that reduce pipeline bubbles while adhering to strict memory budgets. Our approach complements existing offloading techniques: whereas prior approaches trade memory for time in a fixed pattern, we dynamically optimize the tradeoff with respect to model structure and hardware configuration. Experimental results demonstrate that our method consistently improves both throughput and memory utilization. In particular, we reduce idle pipeline time by up to 50% under the same per-device memory limit, and in some cases, enable the training of larger models within limited memory budgets.

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arXiv2025

PDHCG: A Scalable First-Order Method for Large-Scale Competitive Market Equilibrium Computation

Large-scale competitive market equilibrium problems arise in a wide range of important applications, including economic decision-making and intelligent manufacturing. Traditional solution methods, such as interior-point algorithms and certain projection-based approaches, often fail to scale effectively to large problem instances. In this paper, we propose an efficient computational framework that integrates the primal-dual hybrid conjugate gradient (PDHCG) algorithm with GPU-based parallel computing to solve large-scale Fisher market equilibrium problems. By exploiting the underlying mathematical structure of the problem, we establish a theoretical guarantee of linear convergence for the proposed algorithm. Furthermore, the proposed framework can be extended to solve large-scale Arrow-Debreu market equilibrium problems through a fixed-point iteration scheme. Extensive numerical experiments conducted on GPU platforms demonstrate substantial improvements in computational efficiency, significantly expanding the practical solvable scale and applicability of market equilibrium models.

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arXiv2025

Data-driven Mixed Integer Optimization through Probabilistic Multi-variable Branching

In this paper, we propose a Pre-trained Mixed Integer Optimization framework (PreMIO) that accelerates online mixed integer program (MIP) solving with offline datasets and machine learning models. Our method is based on a data-driven multi-variable cardinality branching procedure that splits the MIP feasible region using hyperplanes chosen by the concentration inequalities. Unlike most previous ML+MIP approaches that either require complicated implementation or suffer from a lack of theoretical justification, our method is simple, flexible, provable, and explainable. Numerical experiments on both classical OR benchmark datasets and real-life instances validate the efficiency of our proposed method.

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arXiv2024

Differential Privacy via Distributionally Robust Optimization

In recent years, differential privacy has emerged as the de facto standard for sharing statistics of datasets while limiting the disclosure of private information about the involved individuals. This is achieved by randomly perturbing the statistics to be published, which in turn leads to a privacy-accuracy trade-off: larger perturbations provide stronger privacy guarantees, but they result in less accurate statistics that offer lower utility to the recipients. Of particular interest are therefore optimal mechanisms that provide the highest accuracy for a pre-selected level of privacy. To date, work in this area has focused on specifying families of perturbations a priori and subsequently proving their asymptotic and/or best-in-class optimality. In this paper, we develop a class of mechanisms that enjoy non-asymptotic and unconditional optimality guarantees. To this end, we formulate the mechanism design problem as an infinite-dimensional distributionally robust optimization problem. We show that the problem affords a strong dual, and we exploit this duality to develop converging hierarchies of finite-dimensional upper and lower bounding problems. Our upper (primal) bounds correspond to implementable perturbations whose suboptimality can be bounded by our lower (dual) bounds. Both bounding problems can be solved within seconds via cutting plane techniques that exploit the inherent problem structure. Our numerical experiments demonstrate that our perturbations can outperform the previously best results from the literature on artificial as well as standard benchmark problems.

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arXiv2024

When Does Primal Interior Point Method Beat Primal-dual in Linear Optimization?

The primal-dual interior point method (IPM) is widely regarded as the most efficient IPM variant for linear optimization. In this paper, we demonstrate that the improved stability of the pure primal IPM can allow speedups relative to a primal-dual solver, particularly as the IPM approaches convergence. The stability of the primal scaling matrix makes it possible to accelerate each primal IPM step using fast preconditioned iterative solvers for the normal equations. Crucially, we identify properties of the central path that make it possible to stabilize the normal equations. Experiments on benchmark datasets demonstrate the efficiency of primal IPM and showcase its potential for practical applications in linear optimization and beyond.

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arXiv2024

Orientation Determination of Cryo-EM Images Using Block Stochastic Riemannian Subgradient Methods

The determination of molecular orientations is crucial for the three-dimensional reconstruction of Cryo-EM images. Traditionally addressed using the common-line method, this challenge is reformulated as a self-consistency error minimization problem constrained to rotation groups. In this paper, we consider the least-squared deviation (LUD) formulation and employ a Riemannian subgradient method to effectively solve the orientation determination problem. To enhance computational efficiency, a block stochastic version of the method is proposed, and its convergence properties are rigorously established. Extensive numerical evaluations reveal that our method not only achieves accuracy comparable to that of state-of-the-art methods but also delivers an average 20-fold speedup. Additionally, we implement a modified formulation and algorithm specifically designed to address scenarios characterized by very low SNR.

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arXiv2024

A Robust Anchor-based Method for Multi-Camera Pedestrian Localization

This paper addresses the problem of vision-based pedestrian localization, which estimates a pedestrian's location using images and camera parameters. In practice, however, calibrated camera parameters often deviate from the ground truth, leading to inaccuracies in localization. To address this issue, we propose an anchor-based method that leverages fixed-position anchors to reduce the impact of camera parameter errors. We provide a theoretical analysis that demonstrates the robustness of our approach. Experiments conducted on simulated, real-world, and public datasets show that our method significantly improves localization accuracy and remains resilient to noise in camera parameters, compared to methods without anchors.

cs.CV↗

arXiv2024

Restarted Primal-Dual Hybrid Conjugate Gradient Method for Large-Scale Quadratic Programming

Convex quadratic programming (QP) is an essential class of optimization problems with broad applications across various fields. Traditional QP solvers, typically based on simplex or barrier methods, face significant scalability challenges. In response to these limitations, recent research has shifted towards matrix-free first-order methods to enhance scalability in QP. Among these, the restarted accelerated primal-dual hybrid gradient (rAPDHG) method, proposed by Lu, has gained notable attention due to its linear convergence rate to an optimal solution and its straightforward implementation on Graphics Processing Units (GPUs). Building on this framework, this paper introduces a restarted primal-dual hybrid conjugate gradient (PDHCG) method, which incorporates conjugate gradient (CG) techniques to address the primal subproblems inexactly. We demonstrate that PDHCG maintains a linear convergence rate with an improved convergence constant and is also straightforward to implement on GPUs. Extensive numerical experiments on both synthetic and real-world datasets demonstrate that our method significantly reduces the number of iterations required to achieve the desired accuracy compared to rAPDHG. Additionally, the GPU implementation of our method achieves state-of-the-art performance on large-scale problems. In most large-scale scenarios, our method is approximately 5 times faster than rAPDHG and about 100 times faster than other existing methods. These results highlight the substantial potential of the proposed PDHCG method to greatly improve both the efficiency and scalability of solving complex quadratic programming challenges.

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arXiv2024

A Low-Rank ADMM Splitting Approach for Semidefinite Programming

We introduce a new first-order method for solving general semidefinite programming problems, based on the alternating direction method of multipliers (ADMM) and a matrix-splitting technique. Our algorithm has an advantage over the Burer-Monteiro approach as it only involves much easier quadratically regularized subproblems in each iteration. For a linear objective, the subproblems are well-conditioned quadratic programs that can be efficiently solved by the standard conjugate gradient method. We show that the ADMM algorithm achieves sublinear or linear convergence rates to the KKT solutions under different conditions. Building on this theoretical development, we present LoRADS, a new solver for linear SDP based on the Low-Rank ADMM Splitting approach. LoRADS incorporates several strategies that significantly increase its efficiency. Firstly, it initiates with a warm-start phase that uses the Burer-Monteiro approach. Moreover, motivated by the SDP low-rank theory [So et al. 2008], LoRADS chooses an initial rank of logarithmic order and then employs a dynamic approach to increase the rank. Numerical experiments indicate that LoRADS exhibits promising performance on various SDP problems. A noteworthy achievement of LoRADS is its successful solving of a matrix completion problem with $15,694,167$ constraints and a matrix variable of size $40,000 \times 40,000$ in $351$ seconds.

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