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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 43 ›› Issue (1): 1-14.doi: 10.3969/j.issn.1005-3085.2026.01.001cstr: 32411.14.cjem.CN61-1269/O1.2026.01.001

• •    下一篇

基于新代数等价变换求解Fisher市场均衡问题的全牛顿步内点算法

迟晓妮1,2,  张  璐1,  刘三阳3,  张所滨4   

  1. 1. 桂林电子科技大学数学与计算科学学院，广西高校数据分析与计算重点实验室，桂林 541004
    2. 桂林电子科技大学，广西应用数学中心，桂林 541004
    3. 西安电子科技大学数学与统计学院，西安 710071
    4. 桂林电子科技大学科学技术发展研究院，桂林 541004
  • 收稿日期:2023-04-18 接受日期:2023-08-21 出版日期:2026-02-15 发布日期:2026-04-15
  • 基金资助:
    国家自然科学基金 (12361064)；广西自然科学基金 (2021GXNSFAA220034)；广西科技计划项目 (桂科AD25069086)；桂林电子科技大学研究生教育创新计划 (2022YCXS148).

A Full-Newton Step Interior-point Algorithm for Solving the Fisher Market Equilibrium Based on New Algebraic Equivalent Transformation

CHI Xiaoni1,2,  ZHANG Lu1,  LIU Sanyang3,  ZHANG Suobin4   

  1. 1. School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004
    2. Center for Applied Mathematics of Guangxi Guilin University of Electronic Technology, Guilin 541004
    3. School of Mathematics and Statistics, Xidian University, Xi'an 710071
    4. Institute of Scientific Research and Development, Guilin University of Electronic Technology, Guilin 541004
  • Received:2023-04-18 Accepted:2023-08-21 Online:2026-02-15 Published:2026-04-15
  • Supported by:
    The National Natural Science Foundation of China (12361064); the Natural Science Foundation of Guangxi (2021GXNSFAA220034); the Science and Technology Project of Guangxi (Guike AD25069086); the Innovation Project of Guilin University of Electronic Technology Graduate Education (2022YCXS148).

摘要：

权互补问题是互补问题的一类重要推广，当权向量为零向量时，该问题就化为互补问题。非零权向量的存在使得权互补问题的理论和算法更为复杂。权互补问题的应用广泛，科学、经济等领域中的一大类均衡问题都可以转化为权互补问题进行求解，比如Fisher市场均衡问题可化为一种斜对称的权互补问题。提出了一种求解Fisher市场均衡问题的线性权互补模型的新全牛顿步内点算法。基于中心方程的新代数等价变换形式，运用核函数$\varphi(t)=t^{2}$计算搜索方向。该核函数首次被用于求解线性权互补问题。算法每次迭代仅使用一个全牛顿步，无需进行线搜索，节省运行内存。证明算法的收敛性及多项式复杂度，最后通过数值算例验证了算法的有效性。

关键词: 线性权互补问题, Fisher市场均衡, 全牛顿步, 内点算法, 核函数, 代数等价变换

Abstract:

The weighted complementarity problem is an important generalization of the complementarity problem. When the weight vector is zero, the problem becomes a complementarity problem. The non-zero weight vector makes the theory and algorithm of weighted complementarity problem more complex. The weighted complementarity problem is widely used, such as a large class of equilibrium problems in science, economy and other fields can be transformed into weighted complementarity problem. For example, Fisher market equilibrium problem can be transformed into a skew-symmetric weighted complementarity problem. In this paper, a new full-Newton step interior-point algorithm for is proposed solving Fisher market equilibrium problem's linear weighted complementarity model. Firstly, based on the new algebraic equivalent transformation of the centering equation, the kernel function $\varphi(t)=t^{2}$ is adopted to seek the search direction. This kernel function is first used to solve linear weighted complementarity problem. Secondly, the algorithm only uses a full-Newton step without line search, which saves running memory. We show that the algorithm is convergent and has polynomial complexity. Finally, numerical examples are given to verify the effectiveness of the algorithm.

Key words: linear weighted complementarity problem, Fisher market equilibrium, full-Newton step, interior-point algorithm, kernel function, algebraic equivalent transformation

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