Optimality Zone Algorithms for Hybrid Systems Computation and Control: From Exponential to Linear Complexity
Bibliographic record
Abstract
Necessary conditions were obtained by Shaikh and Caines (2002, 2003 and 2004) for hybrid optimal control problems (HOCPs) which resulted in a general hybrid maximum principle (HMP); further, a class of efficient, provably convergent hybrid maximum principle (HMP) algorithms were obtained based upon the HMP. The notion of optimality zones (OZs) (2004) was introduced as a theoretical framework enabling the computation of optimal schedules for HOCPs (i.e., discrete state sequences with the associated switching times and states). This paper presents the algorithm HMPZ which fully integrates the prior computation of the OZs into the HMP algorithms. Adding (i) the computational investment in the construction of the OZs for a given HOCP, and (ii) the complexity of the computation of the optimal schedule, optimal switching time and state sequence, and the optimal continuous control input, yields a complexity estimate for the algorithm (HMPZ) which is linear (i.e., O(L)) in the number of switching times L; this is to be compared with the geometric (i.e. O(|Q|L)) growth of a direct combinatoric search over the set of schedules, where Q denotes the discrete state set of the hybrid system
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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 | 0.014 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.007 | 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".