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Life sciences · Preprint
arXiv · August 11, 2026
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This is a mathematical characterization of optimal decision policies in partially observable restart systems, proving that optimal policies exhibit threshold structure in elapsed time under specified cost conditions. The work is theoretical and does not evaluate clinical, experimental, or applied outcomes; it establishes mathematical properties of an algorithmic framework that may inform future applications.
Preprint.
Optimal policies exhibit threshold structure in elapsed time under one-step cost deterioration condition for both discounted and undiscounted cost criteria. When state space is partially ordered and kernel is stochastically monotone, optimal threshold is nonincreasing in the state. For average cost criterion, threshold results established via vanishing discount approach under geometric ergodicity and transient gain domination assumptions.
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This is a theoretical mathematical analysis establishing optimal policy structures in a decision-making framework; it advances understanding of algorithmic properties but does not present empirical evidence, clinical outcomes, or data from real-world applications.
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We study a Restart POMDP (Partially Observable Markov Decision Process) on a general Borel state space, where the controller either lets the hidden state evolve unobserved or restarts the system and observes the new state. Exploiting a sufficient-statistic representation consisting of the last observed state and the elapsed time since restart, we reduce the problem to a fully observed MDP. Under a natural one-step cost deterioration condition, we prove that optimal policies have a threshold structure in the elapsed time for both the discounted and total undiscounted cost criteria. When the state space is partially ordered and the kernel is stochastically monotone, we further show that the optimal threshold is nonincreasing in the state. For the average cost criterion, under additional assumptions of geometric ergodicity and domination of the transient gain, we establish analogous threshold results via the vanishing discount approach, after showing the uniform boundedness of the optimal thresholds and relative value functions.
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