13 problems

A restless multi-armed bandit problem has $N$ projects, of which up to $M$ can be selected at each time, where $1\leqslant M<N$. Each project is modeled as a binary-action Markov d…

Order-optimality conjecture. The order optimality of RCA-M should hold when it is used with any index policy that is order optimal for the rested bandit problem.

PCLI3 threshold and marginal PCL identities. The functions $F$ and $G$ admit càdlàg locally bounded-variation extensions to $\mathbb R$, with $G$ nonincreasing and nonconstant. For…

PCLI2 completion conjecture. The function $m_{\rm reg}$ admits a finite-valued, nondecreasing, continuous extension $m\colon[0,\infty)\to\mathbb R$, with $m(x)\to\infty$ as…

PCLI1 completion conjecture. For every $x\in[0,\infty)$ and $z\ne x$, the limit exists and satisfies $\bar N(x,z)>0$. Consequently, the Abelian limit in the prefix-surplus formula…

Let $f(x,z)$ and $g(x,z)$ denote the average-criterion limiting marginal metrics, and define the diagonal MP index by … The average-criterion analogues of $\textup{(PCLI1)}$ and…

Let $p_{01}$, $\rho$, $\kappa$, and $\beta$ be parameters satisfying … A discounted single-project problem is PCL-indexable when the conditions $\textup{(PCLI1)--(PCLI3)}$ hold on…

Uniform-horizon conjecture. The function $\tau(\epsilon):\mathbb{R}\to\mathbb{N}$ is independent of $N$ and depends only on

The restless bandit problem consists of multiple arms whose states evolve whether or not they are activated, subject to a resource constraint that allows a constant fraction of the…

Non-degeneracy conjecture. Non-degeneracy is also a necessary condition for the existence of a policy satisfying

A lazy restless bandit has a belief state $\pi$ and subsidy $\eta$; let $\rho_0,\rho_1$ denote the relevant state-transition parameters, let $R_0,R_1$ denote the rewards associated…

Constraint-specific explanation conjecture. The difference between the results obtained for the paper's RMAB constraints and those obtained for the more general WCDP constraints is…

Extension conjecture. The formulation of the policy and its asymptotic optimality should extend to (i) multiple actions associated with a state, instead of only an active and a pas…