Table of Content

15 June 2026, Volume 30 Issue 2

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Proximal-based methods can guarantee blunt local minimizer for nonconvex nonsmooth optimization problem

WANG Xiangfeng, ZENG Shangzhi, ZHANG Jin, ZHOU Jinchuan

2026, 30(2):  1-23.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.001

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We propose a general Flexible proxImal-based block-wise First-order Algorithm framework called FIFA for a stochastic composite minimization problem with two nonconvex function components in the objective while only one of them is assumed to be differentiable. Under some per-block Lipschitz-like conditions based on Bregman distance, but without the global Lipschitz continuity of the gradient of the differentiable function, we prove that any accumulation point of the sequence is a stationary point of the model. We further show that the stationarity is the "best" one if the global Lipschitz continuity is additionally assumed, and that it is even the local minimizer for some special cases. Convergence analysis without the global Lipschitz continuity and the enhanced stationarity analysis make our results different from existing results in both the convex and nonconvex contexts.

Fast algorithm for OWL1 norm constrained regression model

MEN Yanchao, LI Xudong

2026, 30(2):  24-44.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.002

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As machine learning algorithms continue to evolve and incorporate more features, effective variable selection has become a critical issue. To control the false discovery rate of regression coefficients in the variable selection, Bogdan et al. proposed the SLOPE model in 2015, which considers a linear regression model regularized by the OWL1 norm. In this paper, we consider a new regression model that uses an OWL1 norm constraint, enabling more flexible control of the false discovery rate through two parameters, $\lambda$ and $\tau$. We designed a fast algorithm that efficiently solves this model by using the dual semismooth Newton based proximal point algorithm (PPDNA). The algorithm's outer layer uses the proximal point algorithm, while the inner layer employs the semismooth Newton method to solve the subproblems efficiently. Meanwhile, the special structure of the generalized Jacobian induced by the projector onto the OWL1 norm ball is exploited to efficiently handle the corresponding Newton linear systems in the inner algorithm. Finally, we demonstrate the effectiveness and robustness of PPDNA by comparing it with two popular algorithms using both simulated data and large-scale real datasets.

Solution to strong partition of 2-balanced regular multipartite tournaments

AI Jiangdong, HE Fankang, LIU Yihang

2026, 30(2):  45-57.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.003

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We call a partition of a $c$-partite tournament into tournaments of order $c$ strong if each tournament is strongly connected. The strong partition number, denoted as $ST(r)$, represents the minimum integer $c'$ such that every regular $r$-balanced $c$-partite tournament has a strong partition for all $c \geq c'$. Figueroa, Montellano-Ballesteros, and Olsen showed the existence of $ST(r)$ for all $r\geq 2$ and proved that $5\leq ST(2)\leq 7$. In this note, we establish that $ST(2)=6$ and we also show the unique $2$-balanced $5$-partite tournament which has no strong partition.

The well-posedness of traffic equilibrium problems

ZENG Jing, XIANG Yao, ZHANG Wenyan

2026, 30(2):  58-68.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.004

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In the study of traffic assignment theory, the development of equilibrium model theory has attracted significant attention. The traffic equilibrium problem aims to determine the flow distribution and various performance metrics in a traffic network by using given O/D pair demand and cost functions, based on established path selection criteria. The core of this methodology lies in the fact that, under a traffic equilibrium state, no participant in the system can improve their travel time or cost by unilaterally changing their path choice, thereby maximizing the efficiency and performance of the traffic system. However, in reality, due to the diversity of traffic participants' behavior and the complexity of traffic networks, achieving a true equilibrium state is often challenging. This paper defines an approximate form of traffic equilibrium ${\varepsilon}$-equilibrium flow and verifies its existence. Additionally, it introduces a type of well-posedness for ${\varepsilon}$-equilibrium flow and establishes sufficient conditions for the validity of this well-posedness, providing new theoretical tools and methods for understanding and solving traffic equilibrium problems.

Research of touring route planning based on spatio-temporal awareness

WANG Feng, HANG Bo, HUANG Jinzhou, XU Degang, ZHANG Zeyu, LIU Jiamou

2026, 30(2):  69-78.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.005

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Tourist route planning is a challenging task that requires considering both temporal and spatial dimensions of tourism data. On one hand, it involves obtaining the spatial distribution of tourist attractions within the scenic area, and on the other hand, it takes into account the tourists' visiting behaviors during their exploration of the area. Therefore, in addition to collecting attribute information of various attractions in the scenic area, tourism route planning also requires a significant amount of data on tourists' visiting behaviors. This paper collects the aforementioned data and extracts the travel behavior information of tourists between attractions, proposing important indicators for measuring tourist travel behavior. Based on a comprehensive consideration of these indicators, the Travel Route Planning algorithm (TRP) is introduced with the aim of significantly reducing travel time between attractions. The experiments yield distinct results and corresponding travel times between scenarios without travel route planning and scenarios with travel route planning. Furthermore, by further optimizing the route planning, a comparison of travel times before and after optimization is obtained. The results indicate that the Travel Route Planning algorithm not only greatly reduces travel time but also provides valuable insights for determining optimal locations for tourist buses to make stops. Compared to three representative path planning algorithms currently available, the algorithm proposed in this paper exhibits significant advantages in response time and average solution quality.

Two RMIL-type conjugate gradient methods with sufficient descent property and applications in image restoration

WU Xiaoyu, SHAO Hu, LIU Pengjie, ZHOU Jincheng

2026, 30(2):  79-92.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.006

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The conjugate gradient method possesses the advantages of lower storage requirement and simplicity to iterate, therefore it has been widely used for solving the large-scale optimization problems. Based on the Rivaie-Mustafa-Ismail-Leong (RMIL) conjugate coefficient, we propose two extended RMIL-type coefficients and establish the corresponding conjugate gradient algorithms for solving unconstrained optimization problems. Under the strong Wolfe line search, we prove that the search direction sequence generated by the first algorithm satisfies the descent property. The global convergence property is established under the normal assumptions. The descending property of the second algorithm is independent of any line search condition. By using the standard Wolfe line search, the global convergence of the second algorithm is obtained. To test the numerical effects of two proposed algorithms, we apply them to solve unconstrained optimization problems and restore the blurred images affected by impulse noise. Compared with some existing conjugate gradient methods, experimental results show that the two proposed algorithms are promising.

The full-Newton step feasible interior-point method for general Fisher market equilibrium problems based on a kernel function

CHI Xiaoni, YANG Yuping, LIU Sanyang, YANG Qili

2026, 30(2):  93-108.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.007

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In this paper, we design and analyze a full-Newton step interior-point method (IPM) for solving the weighted linear complementarity problem (WLCP), which is general optimization of the Fisher market equilibrium problem. As a non-trivial generalization of the complementarity problem (CP), weight complementarity problem (WCP) can be used to model a wide range of equilibrium problems in economics, science and engineering. Since there are nonnegative weight vectors in WCP, the theory and algorithms of WCP are more complicated than CP. In this paper, the IPM for CP is extended to solve WCP. Based on a kernel function, search directions are obtained by applying Newton's method to the equivalent system, which defines the central path. Thus, a full-Newton step feasible IPM for general Fisher market equilibrium problem is proposed. At each iteration we only use full-Newton steps, which avoids the calculation of the step size. Under suitable assumptions, the algorithm is shown to have global convergence and polynomial complexity. Some numerical results are provided to support the practical efficiency of the proposed algorithm.

Research on altruistic equilibria in altruistic strategy-form games

WANG Nengfa, YANG Zhe

2026, 30(2):  109-124.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.008

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In this paper, we assume that the players are altruistic in strategy-form games. Following the idea, we introduce the altruistic games and altruistic generalized games with preference correspondences, and prove the existence theorems of altruistic equilibria. As applications, we obtain the existence theorems of altruistic equilibria for normal-form games and generalized games.

The proper jury theorem of voting theory

HU Yuda

2026, 30(2):  125-136.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.009

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The famous Condorcet jury theorem provides the theoretical foundation for voting theory. Based on this theorem, it is only limited that all voters must have the same probability in their preference for the alternatives, which is impossible to happen in real voting, so in fact it only gives a special case of the general situation. This paper substantively extends the Condorcet jury theorem, and establishes a proper jury theorem for the relationship between the probability of the voting group using the majority preference rule to make a strict preference choice for the alternatives when each voter has its own different preference probabilities for the alternatives. At the same time, some important properties of the group strict preference probability determined by the established theorem are given. Finally, it is also proved that when the number of voters increases infinitely, the group strict preference probability determined by this theorem will tend to its maximum limit value of 1.

The inspection and preventive maintenance policy for a δ-shock model which has two types of failures

GAO Qiaoqiao, ZHANG Junnan

2026, 30(2):  137-148.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.010

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The preventive maintenance strategy of a single component repairable system is studied in this paper. The system is subject to random external shocks during operation. The shock has two types: extreme shock and $\delta$-shock. When the working time reaches to a certain value $T$, the preventive maintenance will be carried out, preventive maintenance restores the state of the system to the state it was in after the last repair. There is a certain probability of delayed repair after the system failure, and the system will be replaced by a completely new one after the $N$th failure.According to the renewal reward theory, an expression for the expected cost per unit time of long-term operation is derived by using the replacement strategy $(T,N)$. Finally the feasibility of the model is verified by numerical examples, and the sensitivity analysis of some parameters is made, which can guide enterprises to carry out preventive maintenance of different systems.

Solving the optimal order quantity for an inventory system with unknown parameter values——Based on stock-dependent demand rate

GUO Zhanbing, ZHANG Yejie

2026, 30(2):  149-158.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.011

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Considering the difficulty in obtaining the accurate parameter values in the inventory models, this paper provides a two-stage ordering strategy for one kind of inventory system, where the demand rate depends on the inventory level. By constructing a dynamic order quantity and analyzing its related properties, this two-stage order strategy realizes the estimation of critical value for control parameter, and finally obtains the optimal order quantity. Moreover, this two-stage ordering strategy not only draws on the simple mathematical form of the classic EOQ model, but also does not require the retailer to know the exact model parameter values in advance. Theoretical analysis and numerical simulation demonstrate the feasibility of this strategy. Sensitivity analysis further provides the impact of the estimation error of the control parameter critical value on the effectiveness of this strategy, and the results show that this two-stage ordering strategy is robust to the misestimation of critical value for control parameter.

Pseudo-E-convex functions without differentiability condition and optimality of pseudo-E-convex programming

HUANG Yingquan, JIANG Xuyu, SONG Guihua, YANG Han

2026, 30(2):  159-168.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.012

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Firstly, a class of pseudo-E-convex functions without differentiability condition is defined in this paper, which is a proper generalization of E-convex functions and pseudoconvex functions without differentiability condition. The existence of such functions is verified with an example. Furthermore, the relationships between pseudo-E- convex functions and other functions are discussed. These indicate that pseudo-E-convex functions without differentiability condition are more general and have a wider range of applications. Secondly, some properties of pseudo-E-convex functions are obtained. Finally, in terms of optimality, the relationship between the fixed points of the mapping E and their global optimal solutions is discussed for the pseudo-E-convex programming problems ($\mathrm{P}$) and ($\mathrm{P}_{E}$), the local-global property of the optimal solutions and the uniqueness of the global optimal solution for programming problem ($\mathrm{P}$) are obtained, and the examples are provided to verify the results respectively.

New characterizations of the two-step Shapley-solidarity value and its application

YUAN Meng, LIU Tao, SHAN Erfang

2026, 30(2):  169-178.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.013

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In 2022, Zou et al. proposed a two-step Shapley-solidarity value for cooperative games with coalition structure, which distributes the total worth in two steps. Firstly, players within one union obtain the solidarity value in the subgame restricted to the corresponding union. Then, the surplus of the difference of between the Shapley value of the union obtained in the quotient game, and the worth of the union, will also be allocated to the players in the same union equally. This research proposes a new axiom called the grand coalition solidarity property, and proves that the two-step Shapley-solidarity value can be uniquely characterized by four axioms: efficiency, coalitional balanced contributions, population solidarity within unions and grand coalition solidarity property. Besides, this paper shows that the two-step Shapley-solidarity value is the only value that satisfies efficiency, coalitional symmetry, coalitional marginality, population solidarity within unions and grand union solidarity property. Finally, comparing the two-step Shapley-solidarity value with other values by a numerical example, this paper shows that the two-step Shapley-solidarity value can take care of the weak players better, while ensuring the fair allocation within unions. It turns out that the two-step Shapley-solidarity value reflects a higher degree of solidarity than those values.

An adaptive algorithm for solving quasimonotone variational inequality problems

YU Sijie, LONG Xianjun

2026, 30(2):  179-193.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.014

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In this paper, we propose a new adaptive forward-backward algorithm to solve the quasimonotone variational inequality problem in real Hilbert spaces. Under reasonable assumptions, we prove that the iterative sequence generated by the algorithm converges strongly to an element of the solution set of the variational inequality problem. Finally, numerical experiments show the effectiveness and superiority of the new algorithm.

Inexact proximal point algorithms and projection methods for monotone variational inequalities

CUI Hengxin, JIANG Fan

2026, 30(2):  194-208.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.015

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n this paper, we propose a class of inexact proximal point algorithms with relative error criterion for solving monotone variational inequalities. The next iterate in the proposed methods can be obtained in two ways. Under general hypothetical conditions, the global convergence of the new algorithms is established. By choosing a special form for the error, the proposed inexact proximal point algorithms reduce to a class of projection and contraction methods with linesearch, which reveals the connection between inexact proximal point algorithms and a class of projection methods. Numerical experiments demonstrate the efficiency of the new methods.

Improved q-trust region algorithm for unconstrained optimization problems

QIU Yingming, PENG Jianwen

2026, 30(2):  209-224.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.016

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In this paper, we propose an improved $q$-trust region algorithm for solving unconstrained optimization problems. The algorithm has a new rule for updating the radius of the trust region. We establish the convergence of the improved $q$-trust region algorithm for solving unconstrained optimization problems under the conditions that the function is continuously $q$-differentiable and so on. Finally, numerical experiments show that our algorithm is effective. Compared with the improved $q$-trust region algorithm proposed by Zhou, our proposed improved $q$-trust region algorithm not only iterates to the optimal point faster, but also solves optimization problems with multiple optimal solutions. The results obtained in this paper extend and improve some existing results in the literature.

A sufficient condition for hamiltonicity in t-tough graphs

CHEN Tao

2026, 30(2):  225-231.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.017

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Let $t $ be a nonnegative real number, $S$ be a subset of $V(G)$ and $c(G-S)$ denote the number of components of $G-S$. The graph $G$ is said to be $t$-tough if $|S|\geq t\cdot c(G-S)$ with $c(G-S)\geq 2$ for each vertex set $S$. The toughness is the largest real number $t$ satisfying the above condition. The following result will be proved in this paper. Let $G$ be a $t$-tough graph on $n\geq 3$ vertices with $t\geq 1$. If it holds that $\max \{d(u),d(v)\}>\frac{n}{1+t}+2t-2$ for any two nonadjacent vertices, then $G$ is Hamiltonian.

Edge colorings of planar graphs without 5-fans

XUE Ling, WU Jianliang

2026, 30(2):  232-236.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.018

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A $k$-edge-coloring of a graph is an assignment of colors from a set of $k$ colors to the edges of $G$ such that adjacent edges receive distinct colors. $\chi'(G)$ denotes the smallest $k$ for which $G$ admits such a coloring. It is proved here that if a planar graph $G$ contains no $5$-fan $F_5$ as a subgraph, where $F_5$ is a graph of order $5$ with a vertex $v\in V(F_5)$ such that $d(v)=4$ and $F_5-v$ is a path, then $\chi'(G) \leq \max\{6, \Delta(G)\}$.

A review of intelligent branch and bound algorithms for mixed integer linear programming problems

ZHANG Xuefeng, PENG Xiao, CHEN Liangyu, YANG Zhengfeng, ZENG Zhenbing

2026, 30(2):  237-270.  doi:10.15960/j.cnki.issn.1007-6093.2026.02.019

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Mixed integer linear programming problems are spread across various fields in the real world. Solving mixed integer linear programming problems is an NP-hard problem. Current advanced solvers generally use the branch and bound method as the core framework for solving mixed integer linear programming problems. But the inherent exponential nature of branch and bound means that one wrong decision during its execution can double the size of the search tree and fail to improve the search process. Such a complex and data-rich environment, combined with a lack of formal understanding, makes it possible to leverage machine learning techniques to improve branch and bound algorithms. Therefore, combining data-driven machine learning methods with branch and bound algorithms to improve their decision-making processes has received increasing attention. In this article, we first introduce the branch and bound algorithm and analyze the possible decision-making process therein. Afterwards, the research work on integrating machine learning methods into branch and bound algorithms in recent years is mainly analyzed from two aspects: deep learning methods based on behavioral cloning that imitate existing expert strategies and reinforcement learning methods based on the idea of discovering new strategies. Finally, we discuss possible future directions and challenges in combining machine learning with branch and bound algorithms.