Table of Content

15 March 2026, Volume 30 Issue 1

Previous Issue   

A survey on research advances in consensus-based optimization algorithm

WEI Jiazhen, BIAN Wei

2026, 30(1):  1-23.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.001

Asbtract ( 59 )   PDF (811KB) ( 34 )  

References | Related Articles | Metrics

Global optimization problems have widespread applications across various fields such as scientific research, engineering, economics, and artificial intelligence. Consensus-based optimization algorithm is a class of multi-agent, meta-heuristic and derivative-free algorithms. It is designed to solve global nonsmooth and nonconvex optimization problems, while also being conducive to theoretical analysis and algorithm implementation. In this paper, we first introduce the fundamental principles and analytical results of the original algorithm. Subsequently, the latest development of the consensus-based optimization algorithms and their variants are discussed in detail. And the applications in fields such as machine learning and image processing are briefly described. Finally, we explore future research directions across three key areas: theoretical innovation, algorithm design, and application expansion.

Profit allocation model of service-oriented manufacturing hybrid supply chain under uncertain demand

YU Xiaohui, ZHOU Weiqing, WU You

2026, 30(1):  24-40.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.002

Asbtract ( 43 )   PDF (1027KB) ( 14 )  

References | Related Articles | Metrics

New models such as product and service bundling bring more uncertainty to the customer demand of service-oriented manufacturing, while the service-oriented manufacturing hybrid supply chain is a coalition structure cooperation game with uncertain payoff. In order to solve this uncertain coalition structure cooperative game, a new profit distribution mechanism is proposed based on the internal distribution proportion. It allows each supply chain of the hybrid supply chain to distribute the total profit according to the coalition contribution, while within a single supply chain, the coalition profit is dynamically allocated according to a certain distribution coefficient. This solution reflects the uncertainty of demand in the service-oriented manufacturing hybrid supply chain during the establishment period, and accurately describes the impact of market demand fluctuation on cooperation profit. At the same time, it decreases the impact of incomplete information of coalition profit on the division of total profit.

Differential game models for a new energy vehicle closed-loop supply chain under the applications of blockchain

XU Jianteng, MA Keke, BAI Qingguo, ZHANG Yuzhong

2026, 30(1):  41-60.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.003

Asbtract ( 40 )   PDF (818KB) ( 12 )  

References | Related Articles | Metrics

The application of blockchain technology in the new energy vehicle supply chain can to a certain extent solve the problems of low utilization rate and difficult recycling of waste power batteries. In this context, this paper considers the dynamic strategy for a new energy vehicle closed-loop supply chain consisting of one power battery supplier, one general component supplier and one manufacturer. In this system, blockchain technology is adopted to trace the lifecycle process management of the power battery. The stochastic evolution process of the traceability level is described to show the dynamic changes of the power battery. For the three cases that the power battery supplier and the manufacturer are respectively responsible for recycling products and the vertical integration of both parties is implemented, we construct the corresponding leader-follower stochastic differential game models. By solving the feedback equilibrium, we compare the state variables, decision variables and feedback profits of the three models when the system reaches the steady situation. Numerical examples are provided to conduct the performance of the supply chain in the unsteady-state and dynamic environments. These findings provide some guides for the new energy vehicle enterprises to adopt the blockchain technology, recycle the waste power batteries and integrate the supply chain.

Blockchain empowering the digital transformation: Study on the coordination strategies of supply chain

ZHOU Yunxu, YAO Fanjun, GAO Hongwei

2026, 30(1):  61-74.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.004

Asbtract ( 37 )   PDF (724KB) ( 12 )  

References | Related Articles | Metrics

We used differential game to study the coordination problem of supply chain, taking the digital transformation of the supply chain as the starting point, in which the knowledge accumulation of blockchain is the state variable. We solved and compared the quality improvement, blockchain technology investment, the trajectory of knowledge accumulation, market demand and supply chain profit in non-cooperative and cost-sharing scenarios. We also explored the blockchain technology in the supply chain from the perspective of game theory. Combined with numerical simulation, the sensitivity analysis of relevant parameters was carried out. The study found that the cooperation between players can improve the retailer's blockchain technology investment level without affecting the supplier's quality improvement strategy, effectively alleviate the “double marginal effect” of the non-cooperative case, achieve Pareto improvement of supply chain performance, and then improve the social welfare.

Research on diversification portfolio optimization model and method

ZHAO Hongxin, KONG Lingchen

2026, 30(1):  75-92.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.005

Asbtract ( 95 )   PDF (706KB) ( 10 )  

References | Related Articles | Metrics

Portfolio selection is an important topic in the financial field. Under a series of basic assumptions, economist Markowitz established the mean-variance model in 1952. Then, the study of modern portfolio theory began. Effective diversification is the key to reduce risks and increase returns. This paper starts from mean-variance model and reviews the diversified portfolio strategies. We focus on the portfolio optimization model and solution method under regularization. Finally, we briefly introduce some of our recent work and put forward prospects and ideas based on current research hotspots.

The economics of waiting-area entertainment

SUN Ke, WANG Jinting, WANG Zhongbin

2026, 30(1):  93-107.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.006

Asbtract ( 137 )   PDF (861KB) ( 33 )  

References | Related Articles | Metrics

In order to relieve customers' waiting anxiety when queueing, many service providers provide waiting-area entertainment (WAE) by charging a service fee. This measure can effectively reduce customers' waiting cost and the queueing anxiety of them has been also greatly alleviated. By providing this option, it can not only attract more customers to join the system, but also add additional revenue to service providers. So it is widely favored. This paper establishes a queueing game model by focusing on the popular operation mode of “waiting-area entertainment” in the current service industry, and analyzes the impact of this option on the equilibrium behavior of customers and the revenue of service providers theoretically. The following results are derived in this paper. First, the Nash equilibrium joining strategy of customers is explored under observable and unobservable information cases. Second, the revenue-maximizing information disclosure strategy is derived, and it is optimal to reveal (conceal) queue length information to customers when market size is large (small). The revenue-maximizing service fee increases with the market size. Third, although WAE is provided an additional service option for releasing customers' anxiety for waiting, we find that when the market size is large, this option may hurt consumer surplus.

A faster SDP relaxation for two-period financial derivatives liquidation problem

HUANG Yin, LUO Hezhi

2026, 30(1):  108-120.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.007

Asbtract ( 25 )   PDF (613KB) ( 4 )  

References | Related Articles | Metrics

In this paper, we consider a two-period financial derivative liquidation problem without the convexity assumption. Its optimization model is a non-convex quadratic programming problem with a single quadratic constraint and linear constraints, which is NP-hard. We propose a faster new semi-definite programming (SDP) relaxation for this model by making use of its special structure, and estimate the gap between it and the original problem. We also show that it provides a tighter lower bound than the existing SDP relaxation in the literature. Numerical experiments show that this SDP relaxation can fast provide a very tight lower bound for the original problem, thus can provide effective lower bounds in branch-and-bound algorithm to find the global optimal solution of the problem.

A coordinated multi-agent production and transportation scheduling on parallel machines based on auction algorithm

XU Ke, JI Lanping, GONG Hua, LIU Peng, SUN Wenjuan

2026, 30(1):  121-136.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.008

Asbtract ( 41 )   PDF (828KB) ( 10 )  

References | Related Articles | Metrics

Under the background of resource sharing, in order to solve the competition between different customers' jobs for machine resources, the coordinated production and transportation scheduling problem on parallel machines based on multi-agent is studied. Multiple manufacturers put idle, similar machines on a shared platform, forming a production environment of parallel machines. Jobs from multiple customers need to be processed on machines on the shared platform. Multiple customers with their own optimization objectives are regarded as multiple agents. After the job is processed, the distribution of finished products needs to be considered because the location of machine and customers is dispersed.

Strategic analysis in repairable retrial queueing systems with Bernoulli vacations

HAN Yunna, TIAN Ruiling

2026, 30(1):  137-155.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.009

Asbtract ( 39 )   PDF (790KB) ( 5 )  

References | Related Articles | Metrics

This paper studies the M/M/1 constant retrial queueing model with Bernoulli vacations and server unreliability, where the server breaks down at different rates in normal and idle states. The system has no waiting space and service starts immediately if the arriving customer finds the server is idle. Otherwise, if the system is busy, on vacation, and in a breakdown state, the customer decides whether or not to join the orbit based on the different levels of information provided by the system. After each completed service, the system starts to go on vacation or remains available. The system rejects new customers from entering the system in the event of server breakdown. Based on the different levels of information provided by the system, we study the steady-state indicators in the almost unobservable case and fully unobservable case, as well as the equilibrium strategies of customers in both cases based on the reward-cost structure. Finally, we use numerical examples to show that revealing server status information does not increase the social benefit.

Equilibrium analysis of the fluid model with two types of parallel customers and delayed repair

WANG Jing, XU Xiuli

2026, 30(1):  156-170.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.010

Asbtract ( 38 )   PDF (1493KB) ( 7 )  

References | Related Articles | Metrics

In this paper, the fluid model with two types of parallel customers and fault delay repair is economically analyzed. The normal working state, fault delay repair state and fault repair state are alternately carried out. When the fluid reaches the system, the net benefit based on the obtained information is calculated, and then whether to enter the system is determined. In the case of complete feasibility and almost visibility, the fluid equilibrium balking strategies and the social benefit optimal strategy per unit time are discussed respectively. Numerical examples are given to analyze the influence of the arrival rate and service rate on average social benefits per unit time.

Dynamic programming algorithms for single machine supply chain scheduling

CHEN Rongjun, LIU Yongcai, HUANG He, TANG Guochun

2026, 30(1):  171-178.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.011

Asbtract ( 34 )   PDF (586KB) ( 8 )  

References | Related Articles | Metrics

In this paper, an integrated scheduling model of production and distribution operations is studied. In this model, a set of jobs (i.e., customer orders) are first processed on single machine and then delivered in batches to the downstream customers in different regions. The problem is to find a joint schedule of production and distribution such that an objective function that takes into account both production cost and distribution cost is optimized. Production cost is measured by a function of delivery times, namely, the times when the jobs are delivered to the customers. The distribution cost of a delivery shipment consists of a fixed charge and a variable cost proportional to the total distance of the route taken by the shipment. This paper considers different production costs. For the model with the sum of weighted delivery times as production costs, the strong NP-Hardness has been proven and a dynamic programming algorithm is developed for consistency constraints on jobs' processing time and weight. For the model with production costs related to the due date, the NP-Hardness is analyzed and two dynamic programming algorithms are designed.

Single-machine online scheduling with NDP constraint and deterioration

MENG Lanmeng, MA Ran, ZHANG Yuzhong

2026, 30(1):  179-187.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.012

Asbtract ( 35 )   PDF (720KB) ( 8 )  

References | Related Articles | Metrics

This paper studies the single machine online production scheduling with non-delayed processing (NDP) constraint as well as deterioration. Jobs arriving over time on-line are processed non-preemptive on machine. The model of job processing time is $p_{j}=a+b_{j}t (a>0$). Its objective is to minimize the maximum weighted completion time. Firstly, we derive the lower bound of the considered problem is $1+b_{\max}$ by means of adversary method. Then we design a online algorithm with the competitive ratio of $2+b_{\max}$. Moreover, these online models are simulated to verify the effectiveness and correctness of online algorithm.

Approximation algorithm for mixed batch parallel machine scheduling with nested processing set restrictions

WU Hongyi, WANG Dong, WAN Long, LUO Wenchang

2026, 30(1):  188-196.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.013

Asbtract ( 37 )   PDF (591KB) ( 11 )  

References | Related Articles | Metrics

In this paper, we investigate the mixed batch parallel machine scheduling problem in which a set of jobs should be processed on one of the parallel batch machines with nested processing set restrictions. Each job has its processing time and its machine set for processing with these machine sets satisfying the nested processing set restrictions. Each machine can process a group of jobs as a batch simultaneously, as long as the total number of jobs in this batch does not exceed the capacity of the machine. For a given batch, its processing time is equal to the weighted sum of the maximum processing time and the total processing time of jobs in the batch. The objective function is to minimize the makespan. The problem includes the classic parallel machine scheduling problem as a special case, which is strongly NP-hard. For the studied problem, we derive an approximation algorithm with a performance ratio of $\left({2 + \alpha} \right)$, where $\alpha$ is a given parameter for weight with $0\leq\alpha\leq 1$.

A variance reduced gradient descent ascent algorithm for a class of nonconvex-nonconcave minimax problems

WANG Ziqi, WANG Junlin, XU Zi

2026, 30(1):  197-206.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.014

Asbtract ( 32 )   PDF (543KB) ( 7 )  

References | Related Articles | Metrics

In this paper, we consider a class of stochastic nonconvex-nonconcave minimax problems, i.e., NC-PL minimax problems, for which we assume that the objective function satisfies the Polyak-Łojasiewicz (PL) condition with respect to the inner variable. We propose a variance reduced gradient descent ascent (VRGDA) algorithm for solving NC-PL minimax problems under the stochastic setting. The number of iterations to obtain an $\varepsilon$-stationary point of the VRGDA algorithm for solving NC-PL minimax problems is upper bounded by $\mathcal{O}(\varepsilon^{-3})$. The VRGDA algorithm owns the best iteration complexity in first-order algorithms for solving stochastic NC-PL problems.

A modified conjugate gradient algorithm with its applications in image recovery problems

LIU Cong, JIAN Ailun, YUAN Gonglin

2026, 30(1):  207-216.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.015

Asbtract ( 36 )   PDF (14406KB) ( 8 )  

References | Related Articles | Metrics

It is well-known that, under the WWP (weak Wolfe-Powell) line search technique, the global convergence of the PRP conjugate gradient method for non-convex functions is still open. In this paper, a hybrid conjugate gradient method is proposed for large-scale unconstrained optimization problems. In this method, the modified BFGS method is mixed with the modified PRP conjugate gradient method, and the weak Wolfe-Powell line search technique is used to find the step size, and the search direction has the property of sufficient descent. Theoretically, the global convergence of nonconvex functions is ensured by assuming reasonable conditions. In numerical experiments, the parameter estimation of the Muskingum model reduces the amount of computation and storage, which illustrates the effectiveness of MPRP. In different noise situations, the MPRP is proved to be highly competitive by comparing the recovery of multiple images. In addition, the restoration of the image is more significant under the image of low impulse noise.

Convergence analysis of an adaptive proximal gradient-subgradient algorithm for square-root-loss regression problems

YANG Jinji, SHEN Chungen, YU Zhensheng

2026, 30(1):  217-234.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.016

Asbtract ( 31 )   PDF (793KB) ( 7 )  

References | Related Articles | Metrics

Square-root-loss regression problems have attracted great attention since the choice of its regularization parameter does not rely on the prior knowledge of the deviation of the noise. However, the square-root loss function has a non-differentiable point, which brings difficulties to numerical algorithms. In this paper, we improve proofs of local smoothness and locally restricted strong convexity of the square-root loss function, which is based on the work of Li et al. (2020). To overcome numerical difficulties caused by nonsmoothness of the loss function, we develop an adaptive proximal gradient-subgradient algorithm (APGSA). Under some assumptions, the global convergence of the proposed algorithm is guaranteed with high probability. In addition, we also prove that the algorithm can accurately identify the active manifold in finite iterations, and then the linear rate of local convergence with high probability is established. Finally, simulation experiments were conducted to verify both the effectiveness and the fast local linear convergence of the algorithm APGSA.

The study of the stability of KKMS points

CUI Ruiqi, ZHANG Shu, SONG Qiqing

2026, 30(1):  235-246.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.017

Asbtract ( 27 )   PDF (567KB) ( 5 )  

References | Related Articles | Metrics

KKMS Theorem is the generalization of the famous Sperner Lemma. In the study of existence of many kinds of cores in cooperative game theory, and in the equilibrium analysis of mathematical economics, KKMS Theorem plays an important and fundamental role. Based on the importance of KKMS Theorem in the application of game theory and economics, this paper introduces the conception of KKMS points, constructs the space of KKMS mappings, and studies the stability of the KKMS points, and obtains the semi-continuous and continuous results of KKMS mappings. The results show that KKMS mappings have upper semi-continuity. Further, by constructing a specific counterexample in 2-simplices, this shows that KKMS mappings do not have lower semi-continuity generally. This also gives a sufficient and necessary condition for the continuity of KKMS mappings. Furthermore, this obtains that the KKMS points have generic stability and essential stability, which includes the existing stability results on KKM points as special cases.

Equal division value, equal surplus division value, and differential marginality

YU Zhiqiang, CUI Zeguang, SHAN Erfang

2026, 30(1):  247-255.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.018

Asbtract ( 31 )   PDF (540KB) ( 9 )  

References | Related Articles | Metrics

For cooperative games with transferable utility, equal division value and equal surplus division value are two eminent solutions, and both of them satisfy two standard axioms, additivity and symmetry. To eliminate the controversial additivity, Casajus (2011) proposes differential marginality axiom and explores the relationship between this proposed axiom and both additivity and symmetry. Inspired by Casajus (2011), we employ differential marginality to characterize the equal division value, the equal surplus division value, and their convex combinations.

Isolated toughness variant and the existence of fractional [a,b]-factor

GAO Wei, WANG Weifan

2026, 30(1):  256-266.  doi:10.15960/j.cnki.issn.1007-6093.2026.01.019

Asbtract ( 27 )   PDF (550KB) ( 7 )  

References | Related Articles | Metrics

The existence of fractional factors in specific settings is an important topic of graph factor theory, and isolated toughness is an important parameter to measure the vulnerability of networks. As the unique variant of isolation toughness, $I'(G)$ is defined as the minimum ratio of $|S|$ and $i(G-S)-1$, where $S$ is the subset of vertices that satisfies $i(G-S)\ge2$. This parameter measures the robustness of the network from the perspective of topology, and recent research reveals that it is closely related to the fractional factor. In this paper, we give an $I'(G)$ condition for the existence of fractional $[a,b]$-factors in a graph, and show that the condition is sharp by counterexample. This result extends the original $I'(G)$ tight bound on the existence of the fractional $k$-factor.