Optimality Zone Algorithms for Hybrid Systems Computation and Control: From Exponential to Linear Complexity
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Bibliographic record
Abstract
In [1], [2], [3], [4] necessary conditions were obtained for hybrid optimal control problems (HOCPs) which resulted in a general Hybrid Maximum Principle (HMP); further, in [4], [5], a class of effficient, provably convergent Hybrid Maximum Principle (HMP) algorithms were obtained based up on the HMP. In [3], [4] the notion of optimality zones (OZs) was introduced as a theoretical framework for the computation of optimal location (i.e. discrete state) sequence 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 of [4], [5]. Adding (a) the computational investment in the construction of the OZs for a given HOCP, and (b) the complexity of (i) the computation of the optimal schedule, (ii) the optimal switching time and optimal switching state sequence, and (iii) 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| <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sup> )) growth of a direct combinatoric search over the set of location sequence, where Q denotes the discrete state set of the hybrid system.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it