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

Advancements in research on the immune-inflammatory mechanisms mediated by NLRP3 inflammasome in ischemic stroke and the regulatory role of natural plant products

Ischemic stroke (IS) is a major cause of mortality and disability among adults. Recanalization of blood vessels to facilitate timely reperfusion is the primary clinical approach; however, reperfusion itself may trigger cerebral ischemia-reperfusion injury. Emerging evidence strongly implicates the NLRP3 inflammasome as a potential therapeutic target, playing a key role in cerebral ischemia and reperfusion injury. The aberrant expression and function of NLRP3 inflammasome-mediated inflammation in cerebral ischemia have garnered considerable attention as a recent research focus. Accordingly, this review provides a comprehensive summary of the signaling pathways, pathological mechanisms, and intricate interactions involving NLRP3 inflammasomes in cerebral ischemia-reperfusion injury. Moreover, notable progress has been made in investigating the impact of natural plant products (e.g., Proanthocyanidins, methylliensinine, salidroside, α-asarone, acacia, curcumin, morin, ginsenoside Rd, paeoniflorin, breviscapine, sulforaphane, etc.) on regulating cerebral ischemia and reperfusion by modulating the NLRP3 inflammasome and mitigating the release of inflammatory cytokines. These findings aim to present novel insights that could contribute to the prevention and treatment of cerebral ischemia and reperfusion injury

Mulco: Recognizing Chinese Nested Named Entities Through Multiple Scopes

Nested Named Entity Recognition (NNER) has been a long-term challenge to researchers as an important sub-area of Named Entity Recognition. NNER is where one entity may be part of a longer entity, and this may happen on multiple levels, as the term nested suggests. These nested structures make traditional sequence labeling methods unable to properly recognize all entities. While recent researches focus on designing better recognition methods for NNER in a variety of languages, the Chinese NNER (CNNER) still lacks attention, where a free-for-access, CNNER-specialized benchmark is absent. In this paper, we aim to solve CNNER problems by providing a Chinese dataset and a learning-based model to tackle the issue. To facilitate the research on this task, we release ChiNesE, a CNNER dataset with 20,000 sentences sampled from online passages of multiple domains, containing 117,284 entities failing in 10 categories, where 43.8 percent of those entities are nested. Based on ChiNesE, we propose Mulco, a novel method that can recognize named entities in nested structures through multiple scopes. Each scope use a designed scope-based sequence labeling method, which predicts an anchor and the length of a named entity to recognize it. Experiment results show that Mulco has outperformed several baseline methods with the different recognizing schemes on ChiNesE. We also conduct extensive experiments on ACE2005 Chinese corpus, where Mulco has achieved the best performance compared with the baseline methods

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

Irisin attenuates pyroptosis in high glucose-induced pancreatic beta cells via the miR-133a-3p/FOXO1 axis

Introduction: Irisin is closely related to type 2 diabetes mellitus (T2DM) and other metabolic diseases. It can improve the homeostasis of T2DM. MiR-133a-3p is decreased in the peripheral blood of patients with T2DM. Forkhead box protein O1 (FOXO1) is widely expressed in beta-cells and affects the occurrence of diabetes through transcriptional regulation and signalling pathway regulation. Material and methods: The miR-133a-3p inhibitor was constructed to verify the effect of irisin on pyroptosis through miR-133a-3p. Next, we predicted the presence of targeted binding sequences between FOXO1 and miR-133a-3p by bioinformatics software, which was then confirmed with a double fluorescence assay. Finally, the FOXO1 overexpression vector was used to further verify the effect of irisin through the miR-133a-3p/FOXO1 axis. Results: We first observed that irisin inhibited the protein levels of N-terminal gasdermin D (GSDMD-N) and cleaved caspase-1 and the secretion of interleukins (IL): IL-1beta and IL-18 in Min6 cells treated with high glucoes (HG). Irisin inhibited pyroptosis of Min6 cells treated with HG by reinforcing miR-133a-3p. Then, FOXO1 was validated to be the target gene of miR-133a. Both miR-133a-3p inhibitor and overexpression of FOXO1 restrained the force of irisin on pyroptosis in HG-induced Min6 cells. Conclusion: We explored the protective effect of irisin on HG-induced pyroptosis of islet b-cells in vitro and explained its mechanism of inhibiting pyroptosis through the miR-133a-3p/FOXO1 axis, to provide a theoretical basis for finding new molecular targets to delay beta-cell failure and the treatment of T2DM