A path-following framework on fiber bundle for variational inequalities
Hongbo Sun
Published October 23, 2026 · on the feed October 26, 2026 · ▲ 5 · arXiv 2606.00778
The abstract
Developing general-purpose algorithms with global asymptotic convergence guarantees for variational inequalities (VIs) remains a longstanding challenge. This paper proposes a path-following framework for finite-dimensional VIs with arbitrary continuous mappings and polytope domains. The approach first transforms a general VI into a smooth VI on a simplex. Its key innovation is to formulate the resulting problem on a fiber bundle, called the fixed-point bundle, whose geometric structure enables the systematic integration of initialization, path-following, and singularity avoidance. Without imposing monotonicity or similar assumptions, the framework guarantees global linear convergence for nonsingular solutions and global linear convergence up to any prescribed precision for singular solutions. In numerical experiments, the algorithm attained a precision of 10^{-5} on all 6400 randomly generated instances, with dimensions up to 1024. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.
The Brief
1:47A short podcast summary of the paper. Written and voiced by AI.
Transcript
A path-following framework on fiber bundle for variational inequalities. By Hongbo Sun. Published on the twenty-third of October. Its code repository is public but slight: seven source files.
Variational inequalities have resisted a particular kind of algorithm for a long time. The goal is a general-purpose method, one that is not tailored to a special class of problem, which still comes with a guarantee of global asymptotic convergence. Building that has been a longstanding challenge. The setting here is finite-dimensional: the mapping may be any continuous one, and the domain is a polytope.
The paper proposes a path-following framework for exactly that setting, and its guarantee comes in two parts. For nonsingular solutions, convergence is global and linear. For singular solutions, convergence is still global and linear, but only up to whatever precision is prescribed in advance.
The route there has two moves. First, a general variational inequality is transformed into a smooth one posed on a simplex. Second, and this is the key innovation, the resulting problem is formulated on a fiber bundle, which the author calls the fixed-point bundle. Its geometric structure is what lets three jobs that usually live apart be handled inside one framework: choosing where to start, following the path, and steering clear of singularities.
What the guarantee does not lean on matters as much as what it delivers. No monotonicity is imposed, and no similar assumption takes its place. The requirements stated are a continuous mapping and a polytope domain, and the one concession is the precision limit on singular solutions.
On the numerical side, the algorithm reached a precision of ten to the minus five on every one of six thousand four hundred randomly generated instances, with dimensions as high as one thousand and twenty-four. A reader is left with the question of how far that geometry carries beyond polytopes, and the code and data are released for anyone who wants to probe it.
More with an account
Listen to every day’s papers
Daily playlists, deep dives, quests for the papers, and quizzes on their key terms.
Join free