A belief-state restless bandit model for treatment adherence: Whittle indexability via partial conservation laws
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912817297227776 |
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| author | Niño-Mora, José García, Ángel Pellitero |
| author_facet | Niño-Mora, José García, Ángel Pellitero |
| contents | We study capacity-constrained treatment-adherence outreach via a belief-state restless multi-armed bandit model where patients are a partially observed two-state (adherent/nonadherent) Markov processes and interventions induce reset-type belief dynamics. Using partial conservation laws (PCLs), we establish Whittle indexability of the single-patient problem and derive a closed-form Whittle (marginal productivity) index, together with closed-form reward/work performance metrics under threshold policies and an explicit optimal threshold map. This yields an analytic Lagrangian relaxation: the single-patient Lagrangian value is a piecewise-affine convex function of the intervention price, enabling efficient computation of multi-patient dual bounds and certified relative optimality gaps. We also analyze how the Whittle index depends on the lapse and spontaneous-recovery parameters, providing qualitative insights on intervention priorities. In a large-scale numerical study over heterogeneous two-type populations, we compare Whittle's index policy with a myopic index rule and simple baselines; while myopic is highly competitive on most instances, Whittle's policy yields substantial gains in tight-capacity regimes with a fragile minority, reaching up to about $26\%$ higher reward and markedly smaller relative optimality gaps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_06976 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A belief-state restless bandit model for treatment adherence: Whittle indexability via partial conservation laws Niño-Mora, José García, Ángel Pellitero Optimization and Control 90C40 (Primary) 90C46, 90B36 (Secondary) We study capacity-constrained treatment-adherence outreach via a belief-state restless multi-armed bandit model where patients are a partially observed two-state (adherent/nonadherent) Markov processes and interventions induce reset-type belief dynamics. Using partial conservation laws (PCLs), we establish Whittle indexability of the single-patient problem and derive a closed-form Whittle (marginal productivity) index, together with closed-form reward/work performance metrics under threshold policies and an explicit optimal threshold map. This yields an analytic Lagrangian relaxation: the single-patient Lagrangian value is a piecewise-affine convex function of the intervention price, enabling efficient computation of multi-patient dual bounds and certified relative optimality gaps. We also analyze how the Whittle index depends on the lapse and spontaneous-recovery parameters, providing qualitative insights on intervention priorities. In a large-scale numerical study over heterogeneous two-type populations, we compare Whittle's index policy with a myopic index rule and simple baselines; while myopic is highly competitive on most instances, Whittle's policy yields substantial gains in tight-capacity regimes with a fragile minority, reaching up to about $26\%$ higher reward and markedly smaller relative optimality gaps. |
| title | A belief-state restless bandit model for treatment adherence: Whittle indexability via partial conservation laws |
| topic | Optimization and Control 90C40 (Primary) 90C46, 90B36 (Secondary) |
| url | https://arxiv.org/abs/2601.06976 |