A Practical and Optimal First-Order Method for Large-Scale Convex Quadratic Programming

Convex quadratic programming (QP) is an important class of optimization problem with wide applications in practice. The classic QP solvers are based on either simplex or barrier method, both of which suffer from the scalability issue because their computational bottleneck is solving linear equations. In this paper, we design and analyze a first-order method called the restarted accelerated primal-dual hybrid gradient method for QP, whose computational bottleneck is matrix-vector multiplication. We show that the proposed algorithm has a linear convergence rate when solving generic QP, and the obtained linear rate is optimal among a wide class of primal-dual methods. Furthermore, we connect the linear rate with a sharpness constant of the KKT system of QP, which is a standard quantity to measure the hardness of a continuous optimization problem. Numerical experiments on a standard QP benchmark set showcase the advantage of the proposed algorithm compared to its first-order counterparts

On the Infimal Sub-differential Size of Primal-Dual Hybrid Gradient Method and Beyond

Primal-dual hybrid gradient method (PDHG, a.k.a. Chambolle and Pock method) is a well-studied algorithm for minimax optimization problems with a bilinear interaction term. Recently, PDHG is used as the base algorithm for a new LP solver PDLP that aims to solve large LP instances by taking advantage of modern computing resources, such as GPU and distributed system. Most of the previous convergence results of PDHG are either on duality gap or on distance to the optimal solution set, which are usually hard to compute during the solving process. In this paper, we propose a new progress metric for analyzing PDHG, which we dub infimal sub-differential size (IDS), by utilizing the geometry of PDHG iterates. IDS is a natural extension of the gradient norm of smooth problems to non-smooth problems, and it is tied with KKT error in the case of LP. Compared to traditional progress metrics for PDHG, IDS always has a finite value and can be computed only using information of the current solution. We show that IDS monotonically decays, and it has an $\mathcal O(\frac{1}{k})$ sublinear rate for solving convex-concave primal-dual problems, and it has a linear convergence rate if the problem further satisfies a regularity condition that is satisfied by applications such as linear programming, quadratic programming, TV-denoising model, etc. The simplicity of our analysis and the monotonic decay of IDS suggest that IDS is a natural progress metric to analyze PDHG. As a by-product of our analysis, we show that the primal-dual gap has $\mathcal O(\frac{1}{\sqrt{k}})$ convergence rate for the last iteration of PDHG for convex-concave problems. The analysis and results on PDHG can be directly generalized to other primal-dual algorithms, for example, proximal point method (PPM), alternating direction method of multipliers (ADMM) and linearized alternating direction method of multipliers (l-ADMM)

On the Geometry and Refined Rate of Primal-Dual Hybrid Gradient for Linear Programming

We study the convergence behaviors of primal-dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does

Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming

There is a recent interest on first-order methods for linear programming (LP). In this paper,we propose a stochastic algorithm using variance reduction and restarts for solving sharp primal-dual problems such as LP. We show that the proposed stochastic method exhibits a linear convergence rate for solving sharp instances with a high probability. In addition, we propose an efficient coordinate-based stochastic oracle for unconstrained bilinear problems, which has $\mathcal O(1)$ per iteration cost and improves the complexity of the existing deterministic and stochastic algorithms. Finally, we show that the obtained linear convergence rate is nearly optimal (upto $\log$ terms) for a wide class of stochastic primal dual methods

A feature fusion method using WPD-SVD and t-SNE for gearbox fault diagnosis

The vibration signals of a gearbox always contain the dynamic operation information, which are important for the feature extraction and further work. However, the low signal-to-noise ratio and combined multi-mode faults make it difficult to extract discriminable features of gearboxes. In this study, a feature fusion method based on wavelet packet decomposition (WPD), singular value decomposition (SVD) and t-Distributed stochastic neighbor embedding (t-SNE) for gearbox fault diagnosis is proposed. First, time-frequency analysis method of WPT-SVD as well as time-domain analysis methods are utilized to extract robust feature vectors of gearboxes with different conditions. As an effective method for the visualization of high-dimensional datasets, t-SNE is then introduced to realize the dimensionality reduction of feature vectors. Finally, with the fused features, a radial basis function (RBF) neural network is trained to realize the classification of gearbox fault modes. Sufficient experiments have been implemented to validate the effectiveness and superiority of the proposed method by analyzing the vibration signals of gearboxes

Development and validation of a web-based predictive model for preoperative diagnosis of localized colorectal cancer and colorectal adenoma

BackgroundLocalized colorectal cancer (LCC) has obscure clinical signs, which are difficult to distinguish from colorectal adenoma (CA). This study aimed to develop and validate a web-based predictive model for preoperative diagnosis of LCC and CA.MethodsWe conducted a retrospective study that included data from 500 patients with LCC and 980 patients with CA who were admitted to Dongyang People’s Hospital between November 2012 and June 2022. Patients were randomly divided into the training (n=1036) and validation (n=444) cohorts. Univariate logistic regression, least absolute shrinkage and selection operator regression, and multivariate logistic regression were used to select the variables for predictive models. The area under the curve (AUC), calibration curve, decision curve analysis (DCA), and clinical impact curve (CIC) were used to evaluate the performance of the model.ResultsThe web-based predictive model was developed, including nine independent risk factors: age, sex, drinking history, white blood cell count, lymphocyte count, red blood cell distribution width, albumin, carcinoembryonic antigen, and fecal occult blood test. The AUC of the prediction model in the training and validation cohorts was 0.910 (0.892–0.929) and 0.894 (0.862–0.925), respectively. The calibration curve showed good consistency between the outcome predicted by the model and the actual diagnosis. DCA and CIC showed that the predictive model had a good clinical application value.ConclusionThis study first developed a web-based preoperative prediction model, which can discriminate LCC from CA and can be used to quantitatively assess the risks and benefits in clinical practice

cuPDLP-C: A Strengthened Implementation of cuPDLP for Linear Programming by C language

A recent GPU implementation of the Restarted Primal-Dual Hybrid Gradient Method for Linear Programming was proposed in Lu and Yang (2023). Its computational results demonstrate the significant computational advantages of the GPU-based first-order algorithm on certain large-scale problems. The average performance also achieves a level close to commercial solvers for the first time in history. However, due to limitations in experimental hardware and the disadvantage of implementing the algorithm in Julia compared to C language, neither the commercial solver nor cuPDLP reached their maximum efficiency. Therefore, in this report, we have re-implemented and optimized cuPDLP in C language. Utilizing state-of-the-art CPU and GPU hardware, we extensively compare cuPDLP with the best commercial solvers. The experiments further highlight its substantial computational advantages and potential for solving large-scale linear programming problems. We also discuss the profound impact this breakthrough may have on mathematical programming research and the entire operations research community.Comment: fix typos, update numerical result

Analysis of small RNAs revealed differential expressions during pollen and embryo sac development in autotetraploid rice

Protein-protein interaction of meiosis-related genes with the targets predicted by the DEM associated with meiosis. Table S18. 21 nt-phasiRNAs triggered by the miR2118. Table S19. 24 nt-phasiRNAs triggered by the miR2275. Table S20. Overview of 24 nt TEs-siRNAs during pollen and embryo sac development of 02428-4x and 02428-2x. Table S21. Distribution of 24 nt TEs-siRNAs in autotetraploid and diploid rice. Table S22. Differentially expressed 24 nt TEs-siRNAs during pollen and embryo sac development of autotetraploid rice. Table S23. Anther length during pollen development stages in autotetraploid and diploid rice. Table S24. The stem–loop RT primers used in the present study. (XLSX 1012 kb

Genome-wide and pan-genomic analysis reveals rich variants of NBS-LRR genes in a newly developed wild rice line from Oryza alta Swallen

IntroductionOryza alta Swallen is an allotetraploid perennial wild rice and contains CCDD genome, which may harbor favorable genes for the enrichment of genetic resource.MethodsA new wild rice line, Huaye 5, was developed from Oryza alta Swallen in our lab. Whole genome re-sequencing and pan-genomic analysis were employed to analyze its genomic variations and novel genes.Results and DiscussionMore than ten million genomic variations were detected when compared with Asian cultivar. Among the variational genes, 724, 197 and 710 genes coded protein kinase, synthetase and transcription factor, respectively. A total of 353, 131 and 135 variational genes were associated with morphological trait, physiological trait, resistance or tolerance, respectively. A total of 62 were NBS-LRR genes were detected, in which 11 NBS-LRR genes expressed in sheath and mature stem, and 26 expressed in young and mature roots expressed. The pan-genome sequences of wild rice species with CCDD genome were constructed by integrating 8 Oryza alta (OA), 2 Oryza grandiglumis (OG) and 18 Oryza latifolia (OL) accessions. A total of 28 non-reference NBS-LRR genes were revealed, and 7 of which were mainly expressed in mature roots. This research demonstrated rich DNA variation in the Oryza alta Swallen that may provide a new germplasm for rice resistance breeding