Quasi-Newton methods for multiobjective optimization problems: A systematic review
Bibliographic record
Abstract
Quasi-Newton method is one of the most popular methods for solving unconstrained single and multiobjective optimization problems. In a quasi-Newton method, the search direction is computed based on a quadratic model of the objective function, where some approximations replace the true Hessian at each iteration. Several Hessian approximation schemes with an adequate line search technique provided higher-order accuracies in approximating the curvature and made the methods more effective in terms of convergence to solution. Considering the aforementioned reasons, we write a survey on quasi-Newton methods for multiobjective optimization problems. We discuss the development of all the variants of the quasi-Newton method for multiobjective optimization problems, along with some of the advantages and disadvantages of the existing methods. We give a brief discussion about the line search for these variants too. Two cases have been considered for BFGS, Huang BFGS, and self-scaling BFGS multiobjective versions of quasi-Newton methods: one is in the presence of the Armijo line search, and the other is in the absence of any line search. Subsequently, a nonmonotone line search version is also explained for multiobjective optimization problems. Commentary is given on the convergence properties of these methods. The rate of convergence of all these methods is highlighted. To prove the convergence of every method, it is reported that every sequence of points generated from the method converges to a critical point of the multiobjective optimization problem under some mild assumptions. Based on the available numerical data, we provide an unbiased opinion on the effectiveness of quasi-Newton methods for multiobjective optimization problems.
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.004 | 0.005 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".