On Convergence of Heuristics Based on Douglas-Rachford Splitting and ADMM to Minimize Convex Functions over Nonconvex Sets
Bibliographic record
Abstract
Recently, heuristics based on the Douglas-Rachford splitting algorithm and the alternating direction method of multipliers (ADMM) have found empirical success in minimizing convex functions over nonconvex sets, but not much has been done to improve the theoretical understanding of them. In this paper, we investigate convergence of these heuristics. First, we characterize optimal solutions of minimization problems involving convex cost functions over nonconvex constraint sets. We show that these optimal solutions are related to the fixed point set of the underlying nonconvex Douglas-Rachford operator. Next, we establish sufficient conditions under which the Douglas-Rachford splitting heuristic either converges to a point or its cluster points form a nonempty compact connected set. In the case where the heuristic converges to a point, we establish sufficient conditions for that point to be an optimal solution. Then, we discuss how the ADMM heuristic can be constructed from the Douglas-Rachford splitting algorithm. We show that, unlike in the convex case, the algorithms in our nonconvex setup are not equivalent to each other and have a rather involved relationship between them. Finally, we comment on convergence of the ADMM heuristic and compare it with the Douglas-Rachford splitting heuristic.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.013 | 0.038 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".