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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 (3): 397-416.doi: 10.3969/j.issn.1005-3085.2026.03.001cstr: 32411.14.cjem.CN61-1269/O1.2026.03.001

• •    下一篇

变分不等式意义下凸优化分裂收缩算法的统一框架

何炳生   

  1. 南京大学数学学院，南京 210093
  • 收稿日期:2026-01-26 接受日期:2026-03-26 出版日期:2026-04-15 发布日期:2026-08-15

A Unified Framework of Splitting and Contraction Algorithms for Convex Optimization in the Sense of Variational Inequality

HE Bingsheng   

  1. School of Mathematics, Nanjing University, Nanjing 210093
  • Received:2026-01-26 Accepted:2026-03-26 Online:2026-04-15 Published:2026-08-15

摘要：

科学与工程计算中出现的凸优化问题，很多是带线性约束的。引入乘子以后，问题就可以归结为求其拉格朗日函数的鞍点。鞍点，犹如利益冲突双方的平衡点，它的等价数学表达式是变分不等式的解点。基于这种考虑，提出了一个分裂收缩算法的统一框架，这类方法的每次迭代包括预测和校正两部分：先是通过分裂求解一些形式简单的凸优化子问题去实现预测，然后再由校正提供在一定范数意义下向鞍点(变分不等式解点)的集合收缩的新的迭代点。了解这个并不复杂的框架，就能为各种类型的线性约束的凸优化问题，设计出合适的分裂收缩求解方法。

关键词: 凸优化, 变分不等式, 统一框架, 预测–校正

Abstract:

Many convex optimization problems in scientific and engineering computing involve linear constraints. After introducing multipliers, the problem can be reduced to finding the saddle point of its Lagrange function. The saddle point is like a balance point between conflicting interests, and its equivalent mathematical expression is the solution point of the variational inequality. Based on this consideration, we have proposed a unified framework of splitting and contraction  algorithms, which includes two parts in each iteration: prediction and correction. The prediction is achieved by splitting and solving some simple convex optimization subproblems, while the correction ensures that the iteration point contracts towards the set of saddle points (set of solutions of variational inequalities) in a certain norm. Understanding this uncomplicated framework enables us  to design suitable splitting and contraction methods for various types of convex optimization problems with linear constraints.

Key words: convex optimization, variational inequality, unified framework, prediction-correction

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