Table of Content

15 September 2026, Volume 30 Issue 3

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A multivariate imputation algorithm based on embedding space and its application research

ZHANG Shenkai, LI Ting, WAN Pinglan, HAN Deren

2026, 30(3):  1-22.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.001

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To address the challenges of sparsity and missingness in short-term high-dimensional data, this study proposes a multivariate time series imputation algorithm based on the generalized (q, p) embedding space. The proposed algorithm simultaneously extracts the multivariate joint state information from the q preceding and p succeeding time points around the target imputation node, and establishes bidirectional constraints on the intrinsic relationships among variables, thereby achieving missing value imputation and data augmentation for high-dimensional complex data. Experiments on Traditional Chinese Medicine clinical data for hypertension yield two principal findings. First, regarding predictive inference, the time series data augmented by the proposed algorithm yield a 13.0% improvement in the classification accuracy of the "syndrome of intermin-gled phlegm and blood stasis" based on Long Short-Term Memory. Second, regarding missing value imputation, the binary classification accuracies for the diagnosis of phlegm syndrome and blood stasis syndrome reach 84.2% and 77.2%, respectively.

A distributed adaptive gradient tracking algorithm over multi-agent networks

MU Jiaxin, GAO Juan, LIU Xinwei

2026, 30(3):  23-40.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.002

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This paper studies distributed optimization over undirected networks to minimize the average of local objective functions. Inspired by centralized adaptive gradient methods, we propose a novel distributed adaptive gradient tracking algorithm. The proposed algorithm enables each agent to adaptively update its step size using only local gradient information, completely eliminating the need for manual parameter tuning or global information about the objective functions. For smooth and strongly convex local objective functions, we prove that the generated iteration sequence converges linearly to the optimal solution. Numerical experiments on distributed linear regression problems demonstrate that, owing to a superior step-size update mechanism, the proposed algorithm achieves better convergence performance in comparison with existing distributed methods with fixed step sizes and adaptive step sizes.

A bilevel programming model for optimal portfolio liquidation problem

LI Xuepeng, XU Fengmin

2026, 30(3):  41-55.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.003

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To address the limitation of traditional optimal portfolio liquidation models that overlook strategic interactions, this paper develops a bilevel programming model for portfolio liquidation, characterizing the sequential trading game between a leader and a follower. Theoretical analysis explicitly characterizes the follower's optimal strategy, proves that the cash constraint is active at the optimum, and quantifies the leader's additional loss caused by the follower's trading. A hybrid quantum-behaved particle swarm optimization algorithm is designed to solve the nonconvex nested model. Numerical experiments show that a disadvantaged leader tends to preemptively liquidate vulnerable assets to avoid price declines triggered by the follower; ignoring the game mechanism can result in a relative loss of up to 46.6%.

Traffic flow observability in multi-modal transportation networks: a virtual network-based approach with optimal sensor location

ZHUO Yue, LIANG Yongqi, FAN Yinchao, SHAO Feng, SHAO Hu

2026, 30(3):  56-72.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.004

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Based on graph theory and algebraic system theory, this paper establishes a unified mathematical analysis framework for the observability problem of multimodal transportation networks. The structural complexity arising from the heterogeneity of transportation modes and the complex coupling of transfer behaviors is addressed by integrating multilayer network and hypergraph theories, which leads to the proposal of an isomorphic multilayer virtual network modeling method. Through the introduction of virtual nodes and virtual links, a unified topological-algebraic theory integrating both "structural describability" and "state observability" is constructed. This approach consolidates buses, cars, and transfer arcs within a unified topological structure and maps complex networks containing centroid nodes into virtual networks that satisfy flow conservation, thereby providing a unified and computable algebraic foundation for subsequent analysis. Building upon this framework, this paper systematically derives observability determination theorems based on the node and face methods, establishing explicit mathematical relationships between the minimum number of required observations and topological parameters such as the number of nodes, links, and planes. This transforms the complex problem of "determining the minimum number of sensors to deploy" into concise algebraic expressions. Furthermore, a virtual network construction and sensor optimal deployment algorithm is proposed, providing complete answers to the two core questions: "what is the minimum number of sensors required" and "where should they be deployed." This modeling and solution process essentially constitutes a class of combinatorial optimization problems with topological constraints, reflecting the deep integration of graph-theoretic structures and operational research decision paradigms. Numerical examples demonstrate that the proposed theoretical framework provides a complete solution from topological description to state perception for multimodal transportation networks, achieving precise identification of the minimum sensor set and complete observability of the system state. This work provides a theoretical foundation and a practical pathway for enhancing the operational efficiency of multimodal transportation systems.

Distributed algorithms for generalized Nash equilibrium problems based on Peaceman-Rachford splitting

JIN Zhushi, GAN Tao, XU Lingling

2026, 30(3):  73-98.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.005

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This paper studies generalized Nash equilibrium problems with linear coupling constraints and proposes two distributed algorithms based on operator splitting. By introducing local estimates of other players' decisions for each player, the computation of variational generalized Nash equilibria is reformulated as a zero-finding problem of a double-augmented operator. Based on this reformulation, an inertial relaxed Douglas-Rachford splitting algorithm and a distributed algorithm based on Peaceman-Rachford splitting are constructed. Under linear coupling constraints, convex feasible sets, and the strong monotonicity and Lipschitz continuity of the pseudo-gradient mapping, the convergence of the proposed algorithms is established. Numerical experiments verify the effectiveness of the algorithms.

On the convergence of the alternating direction method of multipliers for non-convex optimization

WANG Qingyang, YUAN Zihang, YANG Jiawen, KONG Xiangyu, CHEN Caihua

2026, 30(3):  99-115.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.006

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The alternating direction method of multipliers (ADMM) is an effective and widely used first-order method for solving composite problems. Nonetheless, existing convergence conditions on ADMM in non-convex settings are too restrictive, limiting ADMM's practical applicability. The main contribution of this paper is the relaxation of existing convergence results of the multi-block proximal ADMM (PADMM) for non-convex optimization problems. Whereas existing results require that there exists a smooth objective term and the associated coefficient matrix has full row rank, we only assume the existence of some smooth terms such that the concatenation of their coefficient matrices is full row rank. Under the weakened conditions, we establish the convergence of PADMM and characterize its convergence rate. Furthermore, for more general cases where the relaxed condition does not hold, we propose a method to equivalently reformulate the original problem such that it satisfies the required full row rank condition and PADMM can be applied.

A variance-reduced federated symmetric alternating direction method of multipliers

JIA Zehui, LI Dandan, XIAO Yujiao, YUE Suyun

2026, 30(3):  116-132.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.007

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To address the core challenges of limited communication bandwidth, insufficient computational resources, and data heterogeneity in federated learning, this paper proposes a variance-reduced federated symmetric alternating direction method of multipliers, termed FedSADMM-VR. Built upon the Symmetric ADMM framework, the proposed method introduces a periodic communication strategy to reduce communication overhead, and employs variance-reduced stochastic gradients to inexactly solve linearized subproblems, thereby effectively lowering both the local computational complexity and the overall communication cost. Furthermore, the algorithm performs symmetric dual variable updates during the local update process, which mitigates the information bias induced by dual variable staleness and enhances the stability of the model under non-independent and identically distributed data. Theoretical analysis establishes that, in the non-convex setting, the algorithm achieves a sublinear convergence rate without requiring the bounded gradient dissimilarity assumption. Experimental results on synthetic data and the a9a, phishing, and FEMNIST datasets demonstrate that FedSADMM-VR can maintain effective test accuracy while reducing the number of communication rounds, and exhibits relatively stable convergence behavior under non-independent and identically distributed data settings.

Inertial Bregman proximal DC algorithm for tensor completion

HU Xiangxin, YAN Xihong, LAN Ze, MU Xiaofang, WANG Chuanlong

2026, 30(3):  133-152.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.008

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This paper studies the tensor completion problem based on Tucker rank for high-dimensional data whose entries are prone to missing during acquisition, transmission, and storage. To alleviate the over-penalization of large singular values caused by nuclear norm relaxation and improve the solution efficiency of the difference-of-convex model, an inertial Bregman proximal difference-of-convex algorithm is proposed within the proximal difference-of-convex algorithm framework. The proposed algorithm introduces an inertial extrapolation strategy and replaces the traditional Frobenius norm proximal term with a Bregman-distance-induced proximal term, thereby characterizing the structural information of tensor completion problems more flexibly. On the theoretical side, under a suitable compactness assumption on the level set and the Kurdyka-Łojasiewicz property, the convergence of the sequence generated by the algorithm is proved, and its limit points are shown to be stationary points of the corresponding optimization model. Experiments on brain MRI image recovery show that, compared with the proximal difference-of-convex algorithm, the proposed algorithm achieves higher recovery accuracy; compared with the high-accuracy low-rank tensor completion algorithm, the proposed algorithm has an advantage in computation time.

High-order tensor-based neural fiber orientation estimation with Rician maximum likelihood

ZHONG Pei, CHEN Yannan

2026, 30(3):  153-169.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.009

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Estimating the fiber orientation density function (fODF) is fundamental for resolving crossing fibers in the human brain white matter and for performing fiber tractography. Conventional spherical deconvolution methods based on least-squares fitting and Gaussian noise assumptions are inadequate for handling Rician noise in magnitude MRI images, especially at low signal-to-noise ratio (SNR), where they tend to introduce biases; moreover, the fODF must satisfy a nonnegative constraint. In this paper, we represent the fODF using high-order tensors, and combine sum-of-squares (SOS) polynomials with semidefinite programming (SDP) to convert the nonnegative constraint into a convex constraint. Meanwhile, we establish a Rician maximum likelihood estimation model. For the likelihood term that contains the modified Bessel function, we adopt the expectation-maximization (EM) algorithm to transform the original problem into a convex subproblem with semidefinite constraints in each iteration, and then introduce the alternating direction method of multipliers (ADMM) for efficient solution, thereby proposing the Rician high-order tensor spherical deconvolution (R-HOT-CSD) method. Experiments on synthetic data, phantom data, and in vivo human brain data demonstrate that the proposed method can effectively suppress spurious peaks and improve angular resolution and estimation accuracy under low-SNR conditions.

A new predictor-corrector algorithm framework for solving monotone inclusion problems

LUO Yueying, LIANG Tianzhe, CAI Xingju

2026, 30(3):  170-182.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.010

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This paper proposes a novel predictor-corrector algorithm framework for monotone inclusion problems. Many classes of problems, such as multi-block separable convex optimization problems and saddle point problems, can be reformulated as monotone inclusions, and algorithms can be designed via the proposed framework. The framework not only encompasses the relaxed forward-backward splitting method and the relaxed Davis-Yin splitting method, but also reproduces Algorithm 3.1 and Algorithm 5.1 in Chang et al (2025) by appropriately selecting the predictor steps and correction matrices, thereby simplifying their convergence analysis.

Quantitative statistical robustness for full random two-stage stochastic optimization problems

ZHANG Jinli, WEI Siyao, TONG Xiaojiao, SUN Hailin

2026, 30(3):  183-203.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.011

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Two-stage stochastic optimization models are widely used in modern operational problems such as investment portfolios and transportation management. In two-stage stochastic optimization, the sample average approximation method is commonly used. The existing research generally assumes that the sample is generated by the real distribution, but the actual situation of the sample may be contaminated, then we need to know whether the optimal value of the two-stage stochastic optimization problem is reliable. This paper investigates the quantitative statistical robustness of the sample average approximation of the optimal value for full random two-stage stochastic optimization problems where the second-stage objective function is a linear function, a piecewise linear function, or a class of convex functions. When the objective function is a linear function, based on the stability analysis of the full random stochastic optimization problem, the quantitative statistical robustness of the sample average approximation to the optimal value is studied. When the objective function is a piecewise linear function, the uniform Lipschitz continuity of the optimal value function in the second stage is derived by proving a uniform bound on feasible dual solutions, thus obtaining the quantitative statistical robustness results of the full random two-stage stochastic optimization problem. The method is extended to a class of full random two-stage stochastic optimization problems where the objective function is a convex function, and the quantitative statistical robustness of the sample average approximation to the optimum of the problem is analyzed.

Two-stage drone facility location and scheduling based on robust satisficing

GUO Yanjing, GAO Pan, WU Zhongming

2026, 30(3):  204-218.  doi:10.15960/j.cnki.issn.1007-6093.2026.03.012

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This paper focuses on the drone facility location and scheduling problem under uncertain demand, and proposes a two-stage location-scheduling model based on robust satisficing. The first-stage decisions cover facility siting, drone allocation, and emergency supply stockpiling. The second-stage operations implement supply dispatching per actual demand. The model employs the Wasserstein distance to characterize demand distribution uncertainty, reconstructs the problem into a solvable adaptive robust optimization model via duality theory and affine decision rules, and solves it using a solver. Numerical results demonstrate that, compared with stochastic programming and distributionally robust optimization models, the proposed model offers stronger robustness in facility location and inventory pre-positioning, and effectively cuts post-disaster emergency transportation costs.