Probability

Motivation

Many computations need the trace of a matrix $A$ that is never formed explicitly and can only be applied to vectors: the trace of a matrix function such as $\mathrm{trace}(A^{-1})$ or $\log\det A = \mathrm{trace}(\log A)$ in statistics and lattice QCD, the Frobenius norm $\|B\|_F^2 = \mathrm{trace}(B^TB)$ of an operator, or the number of triangles of a graph. The standard tool is Monte-Carlo estimation, introduced by M. F. Hutchinson (Hutchinson 1989): average $M$ quadratic forms $z_i^TAz_i$ over random sign vectors $z_i$. Each sample costs one matrix–vector product, uses one random bit per entry, and needs only additions and subtractions.

Before Avron and Toledo (2011), only the variance of such estimators had been analysed. A small variance does not say how many samples guarantee a given relative error with a given probability. Avron and Toledo gave the first bounds of this kind for several estimators. This mission covers the bound for Hutchinson's estimator, their Theorem 7.1.

Setting

A Rademacher random variable takes the values $+1$ and $-1$, each with probability $1/2$. Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive semi-definite. Draw $M \ge 1$ independent random vectors $z_1, \ldots, z_M \in \mathbb{R}^n$ whose $Mn$ entries are independent Rademacher variables. Hutchinson's trace estimator is

$$H_M = \frac{1}{M}\sum_{i=1}^{M} z_i^TAz_i .$$

A single sample $z^TAz$ is an unbiased estimator of $\mathrm{trace}(A)$ (Lemma 2.1 of the paper, due to Hutchinson). For symmetric $A$ its variance is $2\bigl(\|A\|_F^2 - \sum_i A_{ii}^2\bigr)$, twice the squared Frobenius mass of $A$ off the diagonal.

Given $\epsilon > 0$ and $\delta \in (0,1)$, a random estimator $T$ is an $(\epsilon,\delta)$-approximator of $\mathrm{trace}(A)$ if

$$\Pr\bigl(|T - \mathrm{trace}(A)| \le \epsilon\,\mathrm{trace}(A)\bigr) \ge 1 - \delta .$$

The rank $\mathrm{rank}(A)$ is the number of nonzero eigenvalues $\lambda_j$ of $A$, counted with multiplicity.

Formalization targets

Goal: Theorem 7.1, sample bound for $H_M$

For every symmetric positive semi-definite $A$, every $0 < \epsilon \le 1/2$ and every $0 < \delta < 1$,

$$M \ge 6\epsilon^{-2}\ln\!\left(\frac{2\,\mathrm{rank}(A)}{\delta}\right) \quad\Longrightarrow\quad H_M \text{ is an } (\epsilon,\delta)\text{-approximator of } \mathrm{trace}(A).$$

The bound depends on $A$ only through its rank. It does not depend on the dimension $n$, on the condition number, or on how the trace is spread over the diagonal.

Milestones

1. Lemma 7.2 (Achlioptas 2001, Lemma 5). For a unit vector $\alpha \in \mathbb{R}^n$ and $S = \frac1M\sum_{i=1}^M(\alpha^Tz_i)^2$, for every $\epsilon > 0$,

$$\Pr(|S - 1| \ge \epsilon) \le 2\exp\!\left(-\frac{M}{2}\left(\frac{\epsilon^2}{2} - \frac{\epsilon^3}{3}\right)\right).$$

2. Per-direction bound (proof of Theorem 7.1, p. 8:11). Let $r \ge 1$ and $0 < \epsilon \le 1/2$. If $M \ge 6\epsilon^{-2}\ln(2r/\delta)$, then $\Pr(|S - 1| \ge \epsilon) \le \delta/r$.
3. From directions to the trace (proof of Theorem 7.1, p. 8:11). Write $A = U\Lambda U^T$ and $y_i = U^Tz_i$. If $|\frac1M\sum_i y_{ij}^2 - 1| \le \epsilon$ for every $j$ with $\lambda_j \ne 0$, then $|H_M - \mathrm{trace}(A)| \le \epsilon\,\mathrm{trace}(A)$. This step is deterministic.
4. Lemma 2.1 (Hutchinson). $\mathrm{E}(z^TAz) = \mathrm{trace}(A)$, and for symmetric $A$, $\mathrm{Var}(z^TAz) = 2(\|A\|_F^2 - \sum_i A_{ii}^2)$.

Significance

Theorem 7.1 gives a practitioner an explicit number of matrix–vector products after which Hutchinson's method is guaranteed to reach relative accuracy $\epsilon$ with confidence $1-\delta$. No bound was available before, although the method had been in wide use for two decades. The bound exceeds the one the same paper proves for Gaussian test vectors ($20\epsilon^{-2}\ln(2/\delta)$) by a $\ln(\mathrm{rank}(A))$ factor. The authors conjecture that this factor is not needed. Later work removed it: Roosta-Khorasani and Ascher (2015) proved a rank-free bound for the Rademacher case, and Cortinovis and Kressner (2022) extended sample bounds to indefinite matrices. Sample bounds of this type underlie variance-reduced estimators such as Hutch++ (Meyer et al. 2021).

The theorem is proved in the literature. To the platform's knowledge it has no machine-checked proof. Formalizing it produces a checked Rademacher concentration inequality for averages of squared linear forms (Achlioptas' lemma, which is also a core lemma of database-friendly Johnson–Lindenstrauss projections), and a checked spectral reduction from a quadratic-form estimator to its eigen-directions. Neither is currently in Mathlib.

Difficulty

The estimator is not an average of independent copies of a bounded variable with a small range: a single sample $z^TAz$ can deviate from $\mathrm{trace}(A)$ by an amount comparable to $\|A\|_F\sqrt n$. Chebyshev's inequality with the variance of Lemma 2.1 only gives a polynomial dependence on $1/\delta$. Hoeffding's inequality applied to $z^TAz$ directly gives a dependence on $n$ and on the size of the entries of $A$. Unlike the Gaussian case, the estimator cannot be written as a weighted sum of independent chi-squared variables, because rotating a Rademacher vector does not give another Rademacher vector. The central difficulty is Lemma 7.2: a tail bound for $(\alpha^Tz)^2$ that holds uniformly over every unit direction $\alpha$, including directions in which $\alpha^Tz$ is far from Gaussian (for $\alpha = e_1$ it is a constant).

Formalization scope

Matrices are Matrix (Fin n) (Fin n) ℝ, and positive semi-definiteness is Matrix.PosSemidef. The Rademacher law is $\tfrac12(\delta_1 + \delta_{-1})$ on ℝ. The sample space is Fin M → Fin n → ℝ with the product of $Mn$ copies of this law (hutchinsonSampleMeasure), so the law of the estimator is constructed, not assumed. $H_M(\omega) = (M:\mathbb{R})^{-1}\sum_i \omega_i\cdot(A\omega_i)$. Probabilities are Measure.real, and the $(\epsilon,\delta)$-approximator is Definition 4.1 verbatim. $\mathrm{rank}(A)$ is Matrix.rank. In Lemma 7.2 the i.i.d. copies $Q_i$ are the functions $(\alpha\cdot z_i)^2$ on this product space.

Corrections of printed statements, each recorded in the item's Formalization Note:

• Theorem 7.1 is printed without a range on $\epsilon$. Its proof needs $\frac{M}{2}(\frac{\epsilon^2}{2} - \frac{\epsilon^3}{3}) \ge \frac{M\epsilon^2}{6}$, which holds exactly when $\epsilon \le 1/2$. Without a range the printed statement is false: take $A = \frac1n\mathbf 1\mathbf 1^T$, $n = 10^4$, $\epsilon = 10$ and $\delta = 10^{-4}$. The condition admits $M = 1$, while $\Pr(H_1 > 11) \approx 9\cdot10^{-4} > \delta$. The goal and milestone 2 are therefore stated for $0 < \epsilon \le 1/2$.
• Lemma 2.1 is printed for an arbitrary $n\times n$ matrix. The variance formula fails for non-symmetric $A$: for $A = \begin{pmatrix}0&1\\0&0\end{pmatrix}$ the variance is $1$, not $2$. Symmetry is assumed for the variance part only.
• Proof of Theorem 7.1. The proof writes $\Lambda = UAU^T$ together with $y_i = U^Tz_i$. These fit together only for $A = U\Lambda U^T$, which is the convention of milestone 3.

For $A = 0$ the threshold involves $\ln 0$. Lean's Real.log 0 = 0 turns the condition into $M \ge 0$, which agrees with the paper's reading $\ln 0 = -\infty$. The conclusion then holds because $H_M = \mathrm{trace}(A) = 0$, so no extra hypothesis is added. A formalization that took the law of the samples as a hypothesis could make that hypothesis unsatisfiable and the theorem vacuous; the constructed product space rules this out.

A complete development needs:

• the product Rademacher measure and moment generating functions of Rademacher sums;
• a Chernoff bound for averages of i.i.d. bounded variables on a product space;
• the spectral theorem for real symmetric matrices, with rank equal to the number of nonzero eigenvalues;
• a finite union bound.

The Rademacher concentration results are reusable beyond this mission. Proofs of any milestone are welcome, as are alternative proofs of Lemma 7.2.

Selected references

• H. Avron and S. Toledo, Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix, Journal of the ACM 58(2), Article 8, 2011. https://doi.org/10.1145/1944345.1944349
• M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines, Communications in Statistics – Simulation and Computation 18(3), 1059–1076, 1989. https://doi.org/10.1080/03610918908812806
• D. Achlioptas, Database-friendly random projections, Proceedings of PODS 2001, 274–281. https://doi.org/10.1145/375551.375608
• F. Roosta-Khorasani and U. Ascher, Improved bounds on sample size for implicit matrix trace estimators, Foundations of Computational Mathematics 15, 1187–1212, 2015. https://doi.org/10.1007/s10208-014-9220-1
• A. Cortinovis and D. Kressner, On randomized trace estimates for indefinite matrices with an application to determinants, Foundations of Computational Mathematics 22, 875–903, 2022. https://doi.org/10.1007/s10208-021-09525-9
• R. A. Meyer, C. Musco, C. Musco and D. P. Woodruff, Hutch++: Optimal stochastic trace estimation, SOSA 2021, 142–155. https://doi.org/10.1137/1.9781611976496.16

Motivation

A Markov Decision Model (MDM) formalizes sequential decision-making under uncertainty: a controller observes the current state of a system, chooses an action, receives a reward, and the system moves to a new (random) state whose law depends on the current state and action. This framework underlies dynamic programming across operations research, economics, and engineering — inventory control, sequential portfolio choice, queueing control, and reinforcement learning are all instances of it. The finite-horizon theory developed here is the foundation on which every later chapter of Bäuerle and Rieder's Markov Decision Processes with Applications to Finance (Springer, 2011) builds, including the infinite-horizon, partially observed, and optimal-stopping variants treated later in the book.

The classical treatment of dynamic programming for finite state and action spaces goes back to Bellman (1957) and is standard textbook material (see e.g. Puterman, Markov Decision Processes, 1994). The generalization to Borel state and action spaces — needed as soon as a state variable is continuous, as in almost every financial application — requires genuine measure-theoretic care: suprema over an infinite (even uncountable) action set need not be attained, and the resulting value function need not be measurable. Bertsekas and Shreve's Stochastic Optimal Control: The Discrete Time Case (1978) is the classical reference for this general theory; Bäuerle and Rieder's treatment isolates the exact abstract hypothesis — the Structure Assumption (SAN) below — under which the finite-horizon theory goes through cleanly, separating the recursive (Bellman) machinery from the case-by-case verification of when it applies.

Setting

A Markov Decision Model with planning horizon $N \in \mathbb{N}$ consists of a state space $E$ and action space $A$ (measurable spaces), and, for each stage $n = 0,\dots,N-1$: a measurable set $D_n \subseteq E \times A$ of admissible state-action pairs (containing the graph of some measurable selection $E \to A$); a stochastic transition kernel $Q_n(\cdot \mid x,a)$ giving the law of the next state; a measurable one-stage reward $r_n : D_n \to \mathbb{R}$; and a terminal reward $g_N : E \to \mathbb{R}$.

A decision rule at time $n$ is a measurable $f_n : E \to A$ with $f_n(x) \in D_n(x) := \{a : (x,a) \in D_n\}$ for every $x$; an $N$-stage policy $\pi = (f_0,\dots,f_{N-1})$ is a sequence of such rules. Given $\pi$ and an initial state $x$ at time $n$, the process evolves as a (non-stationary) Markov chain, and the value of $\pi$ is the expected total reward

$$V_n^\pi(x) := \mathbb{E}^\pi_{n,x}\!\left[\sum_{k=n}^{N-1} r_k\bigl(X_k, f_k(X_k)\bigr) + g_N(X_N)\right],$$

with value function $V_n(x) := \sup_\pi V_n^\pi(x)$, the best attainable expected reward. A policy is optimal if $V_0^\pi = V_0$. Write $\mathrm{IM}(E)$ for the measurable functions $E \to [-\infty,\infty)$ (never $+\infty$), and define, for $v \in \mathrm{IM}(E)$, the one-step operators $L_n v(x,a) := r_n(x,a) + \int v(x')\, Q_n(dx' \mid x,a)$, $T_n^f v(x) := L_n v(x, f(x))$, and $T_n v(x) := \sup_{a \in D_n(x)} L_n v(x,a)$. A decision rule $f$ is a maximizer of $v$ at time $n$ if $T_n^f v = T_n v$.

Formalization targets

Goal: Theorem 2.3.8 (the Structure Theorem)

Under the Structure Assumption (SAN) — the existence of sets $\mathrm{IM}_n \subseteq \mathrm{IM}(E)$, $\Delta_n \subseteq F_n$ with $g_N \in \mathrm{IM}_N$, $T_n$ mapping $\mathrm{IM}_{n+1}$ into $\mathrm{IM}_n$, and every $v \in \mathrm{IM}_{n+1}$ admitting a maximizer in $\Delta_n$ — the value function satisfies the Bellman equation $V_n = T_n V_{n+1}$ (with $V_n \in \mathrm{IM}_n$), and every sequence of maximizers of $V_1,\dots,V_N$ defines an optimal policy. This is the weakest level at which the theorem holds: it names an abstract structural hypothesis rather than a specific sufficient condition (e.g. compactness plus semicontinuity, treated in a later mission), so any future refinement of sufficient conditions for (SAN) leaves this statement untouched.

Significance

Reducing an $N$-stage optimization over an infinite-dimensional policy space to $N$ one-stage optimizations — literally the content of the Bellman equation — is what makes dynamic programming computationally and theoretically tractable at all. For finite state and action spaces this reduction is elementary (a supremum over a finite set is always attained); the content of the Structure Theorem is doing this correctly when $E$, $A$ are general Borel spaces, where existence of the supremum and of a measurable maximizing selection are not automatic and must be assumed abstractly.

Formalizing this theorem produces machine-checked statements of the finite-horizon Bellman equation and verification theorem in the generality actually used throughout the book's finance applications (wealth is real-valued, portfolios are vector-valued — never finite sets). The statement, proof, and every hypothesis are original to this textbook chapter; no formalized version of this general-Borel-space theory exists on the platform. The closest prior art, finite state-and-action-space Bellman equations and verification theorems (e.g. discounted infinite-horizon and stochastic-shortest-path theorems for finite MDPs), is a strictly weaker special case in which the Structure Assumption's existence-of-maximizer clause is automatic; this mission's goal is not restated as a reference to that prior art; the generalization is exactly the mission's content.

Difficulty

The obvious first attempt — prove $V_n = T_n V_{n+1}$ directly from the definitions of $V_n$ and $V_n^\pi$ — runs into two separate obstructions that (SAN) is built to bypass simultaneously. First, $\sup_\pi V_n^\pi(x)$ and $\sup_{a \in D_n(x)} L_n V_{n+1}(x,a)$ are a priori different suprema (over policies versus over actions), and showing they agree requires that the pointwise supremum over decision rules $f \in F_n$ of $L_n V_{n+1}(x, f(x))$ equals the supremum over bare actions $a \in D_n(x)$ — which needs a measurable selection achieving (or approaching) the action-wise optimum, not just its existence pointwise. Second, $V_{n+1}$ itself must be shown measurable — an a priori supremum of measurable functions over an uncountable index set (all policies) need not be measurable — before the integral $\int V_{n+1}\, dQ_n$ even makes sense. (SAN)'s three clauses are exactly what supplies both a well-behaved measurability class $\mathrm{IM}_n$ closed under $T_n$ and a measurable maximizing selection at every stage, letting a backward induction on $n$ establish both facts together.

Formalization scope

State and action spaces are arbitrary measurable spaces (MeasurableSpace E, MeasurableSpace A type classes), not restricted to Borel subsets of Polish spaces, since none of this mission's statements use topology. Time is indexed by ℕ rather than Fin N, with n < N as an explicit side condition throughout (documented in MODERATION_NOTES.md); this changes no content but avoids Fin-cast noise in the several backward recursions the chapter's operators require. Extended-real values (EReal) are used throughout for value functions, restricted by hypothesis to never equal $+\infty$, matching $\mathrm{IM}(E) := \{v : E \to [-\infty,\infty)\}$ exactly; a formalization using plain ℝ-valued value functions would be a strictly stronger — and unfaithful — claim, since it silently assumes no policy can drive the expected reward to $-\infty$.

The mission does not construct the canonical path measure on the full trajectory space via the Ionescu–Tulcea theorem; the value of a policy is instead built as an explicit backward accumulator recursion over the model's one-step kernels, which computes the same quantity by the tower property of conditional expectation. Definitions 2.1.1, 2.1.5, 2.2.2, 2.3.1, and 2.3.6 — the Markov Decision Model, (Markov and history-dependent) policies, and the operators — are restated from scratch in this mission's own namespace, since drafts in this series cannot import one another; later missions in the same series restate the same vocabulary independently. A trivializing formalization of the goal would state $V_n$ as an unspecified object merely postulated to satisfy the Bellman equation (Theorem 2.3.7's weaker claim) rather than as the supremum-over-policies value function fixed before the theorem — this mission states the latter.

Selected references

• N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, Universitext, Springer, 2011. DOI: 10.1007/978-3-642-18324-9.
• D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, 1978.
• K. Hinderer, Foundations of Non-stationary Dynamical Programming with Discrete Time Parameter, Lecture Notes in Operations Research and Mathematical Systems 33, Springer, 1970.
• M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994.

Motivation

Many matrices in scientific computing are available only implicitly: one can multiply a vector by $A$, for instance by solving a linear system or applying a matrix function, but the entries of $A$ are never formed. Quantities such as the trace must then be estimated from a small number of matrix–vector products. Randomized trace estimators do exactly this: draw random vectors $z$, compute the quadratic forms $z^TAz$, and average.

Avron and Toledo (J. ACM 58(2), 2011) gave the first sample bounds of the form "$M$ samples suffice for relative error $\epsilon$ with probability $1-\delta$" for the standard estimators. Among their results is a case in which the estimate is not merely approximate but exact: when $A$ is a projection matrix, its trace equals its rank, an integer, and rounding a Gaussian trace estimate recovers that integer with high probability. Computing the rank of a projection arises, for example, when charge densities are computed in electronic structure calculations without diagonalization (Bekas, Kokiopoulou and Saad 2007).

Setting

Let $n \ge 1$ and $A \in \mathbb{R}^{n\times n}$. A projection matrix here is an orthogonal projection: $A$ is symmetric and $A^2 = A$. Equivalently, $A$ is symmetric and every eigenvalue of $A$ is $0$ or $1$; its rank $\mathrm{rank}(A)$ is the number of eigenvalues equal to $1$.

Fix a number of samples $M \ge 1$. Let $z_1,\ldots,z_M \in \mathbb{R}^n$ be random vectors whose $Mn$ entries are independent standard normal random variables. The Gaussian trace estimator (Definition 3.1 of the paper) is

$$G_M = \frac{1}{M}\sum_{i=1}^{M} z_i^T A z_i .$$

Its expectation is $\mathrm{trace}(A)$. For $x \in \mathbb{R}$, $\mathrm{round}(x)$ denotes the nearest integer to $x$. For $k \ge 1$, a random variable $X$ has the $\chi^2$ distribution with $k$ degrees of freedom, $X \sim \chi^2(k)$, if it has the law of $g_1^2 + \cdots + g_k^2$ for independent standard normal $g_1, \ldots, g_k$.

Formalization targets

Goal: Lemma 5.3 (p. 8:9)

For a projection matrix $A$, a failure probability $\delta > 0$, and every integer $M \ge 1$ with $M \ge 24\,\mathrm{rank}(A)\ln(2/\delta)$,

$$\Pr\bigl(\mathrm{round}(G_M) \ne \mathrm{rank}(A)\bigr) \le \delta .$$

The number of samples depends on the rank and on $\delta$ only; there is no accuracy parameter, and none on $n$.

Milestones, in the order the proof uses them

1. Law of $MG_M$. $MG_M \sim \chi^2(M\,\mathrm{rank}(A))$.
2. $\chi^2$ tail bound (cited by the paper from Li, Hastie and Church 2007). For $X \sim \chi^2(k)$, $k\ge1$ and $0 < \epsilon \le \tfrac12$,

$$\Pr(|X - k| \ge \epsilon k) \le 2\exp(-k\epsilon^2/6).$$

3. Tail of $G_M$. For $\mathrm{rank}(A) \ge 1$ and $0 < \epsilon \le \tfrac12$,

$$\Pr(|G_M - \mathrm{rank}(A)| \ge \mathrm{rank}(A)\epsilon) \le 2\exp(-M\,\mathrm{rank}(A)\epsilon^2/6).$$

4. Eq. (2). If moreover $M \ge 6\,\mathrm{rank}(A)^{-1}\epsilon^{-2}\ln(2/\delta)$, then $\Pr(|G_M - \mathrm{rank}(A)| \ge \mathrm{rank}(A)\epsilon) \le \delta$.
5. Trace of a projection. $\mathrm{trace}(A) = \mathrm{rank}(A)$.

Significance

The lemma turns a randomized estimator into an exact algorithm with a controlled failure probability: the rank of an implicitly given projection is obtained from $O(\mathrm{rank}(A)\log(1/\delta))$ matrix–vector products, independently of the dimension $n$. It is also an instance where the paper's general relative-error bound for the Gaussian estimator (Theorem 5.2, whose sample count grows like $\epsilon^{-2}$) is improved by exploiting the spectrum of $A$: an absolute error below $\tfrac12$ is a relative error $\epsilon = 1/(2\,\mathrm{rank}(A))$, for which the general bound would require a number of samples quadratic in the rank, whereas Lemma 5.3 needs only a linear number.

The result is proved in the paper. The mission produces a machine-checked version of it, together with two pieces of reusable substrate: the exact $\chi^2$ law of a Gaussian quadratic form in an orthogonal projection, and a two-sided $\chi^2$ tail bound with explicit constant $1/6$ on the range $0<\epsilon\le 1/2$. To the best of available knowledge, none of these statements is formalized in Mathlib; a platform mission on the Johnson–Lindenstrauss lemma states a $\chi^2$ concentration bound with a different exponent, $(\epsilon^2-\epsilon^3)/4$, on the open range $0<\epsilon<1/2$, which does not cover the value $\epsilon = 1/2$ needed here when $\mathrm{rank}(A)=1$.

Difficulty

The deterministic part is short; the probabilistic part is not. The step "$y = Uz$ has independent standard normal entries because $U$ is orthogonal" is the rotation invariance of the standard Gaussian measure on $\mathbb{R}^n$, and it must be combined with the independence of the $M$ samples to identify the law of a sum of $M\,\mathrm{rank}(A)$ squares; this is a statement about product measures and pushforwards, not about moments. The $\chi^2$ tail bound is quoted by the paper without proof. Its constant $1/6$ is not the constant of the usual textbook $\chi^2$ estimates, and the bound is false outside a restricted range of $\epsilon$ (see below), so a generic sub-exponential concentration inequality with unspecified constants does not deliver it. Finally, the rounding step requires matching Mathlib's round with the event $|G_M - \mathrm{rank}(A)| < \tfrac12$.

Formalization scope

All declarations live in the namespace TraceEstimation.ProjectionRank. Matrices are Matrix (Fin n) (Fin n) ℝ. The sample space of $G_M$ is Fin M → Fin n → ℝ with the product measure Measure.pi (fun _ => Measure.pi (fun _ => gaussianReal 0 1)), so the law of the estimator is constructed, not assumed. Probabilities are Measure.real of events. "Projection matrix" is A.IsHermitian ∧ A * A = A (orthogonal projection); a non-symmetric idempotent also has eigenvalues $0$ and $1$, but the paper's proof diagonalizes $A$ by a unitary matrix, which requires symmetry. round is Mathlib's round : ℝ → ℤ; the rank is Matrix.rank. The $\chi^2$ distribution is not defined: its role is played by the pushforward of a product of standard normals under $g \mapsto \sum_l g_l^2$.

Corrections of the printed text, both in the $\chi^2$ tail bound (milestone 2):

• The paper prints $\Pr(|X - k| \le \epsilon k) \le 2\exp(-k\epsilon^2/6)$. The inner $\le$ is a misprint for $\ge$; the next display applies the bound with $\ge$.
• The paper gives no range for $\epsilon$. The bound fails for $\epsilon = 1$ and large $k$, since $\Pr(X \ge 2k)$ decays like $e^{-k(1-\ln 2)/2}$, slower than $e^{-k/6}$. The mission states it for $0<\epsilon\le 1/2$, and milestones 3 and 4 inherit that range; the proof uses only $\epsilon = 1/(2\,\mathrm{rank}(A)) \le 1/2$.

The goal keeps the paper's hypotheses: $\delta > 0$ with no upper bound, and $\mathrm{rank}(A) = 0$ allowed (both are true cases). A statement about the event $|G_M - \mathrm{rank}(A)| < \tfrac12$, or Eq. (2) alone, is not the goal; the goal is about $\mathrm{round}(G_M)$. The sample measure is a probability measure, so the bound $\le \delta$ is not vacuous.

Contributions welcome beyond the milestones: rotation invariance of the standard Gaussian vector under orthogonal matrices, the moment generating function of $\chi^2(k)$, and the spectral fact $\mathrm{trace}(A) = \mathrm{rank}(A)$ for idempotents, each reusable well outside this mission.

Selected references

• H. Avron and S. Toledo, Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix, Journal of the ACM 58(2), Article 8, 2011. https://doi.org/10.1145/1944345.1944349
• P. Li, T. Hastie and K. Church, Nonlinear estimators and tail bounds for dimension reduction in $l_1$ using Cauchy random projections, in Learning Theory (COLT 2007), Lecture Notes in Computer Science 4539, Springer, 514–529, 2007 (the version the paper cites); journal version in Journal of Machine Learning Research 8, 2497–2532, 2007, https://jmlr.org/papers/v8/li07b.html
• C. Bekas, E. Kokiopoulou and Y. Saad, An estimator for the diagonal of a matrix, Applied Numerical Mathematics 57(11–12), 1214–1229, 2007. https://doi.org/10.1016/j.apnum.2007.01.003
• M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines, Communications in Statistics – Simulation and Computation 19(2), 433–450, 1990. https://doi.org/10.1080/03610919008812866

Motivation

Mission XII (12-packet-networks-model) built the discrete-time, slotted packet-network model from scratch and proved the chapter's own version of the fluid-to-stochastic stability bridge (Theorem 12.10). That result is only useful once paired with an actual control policy whose fluid model can be shown stable. J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) supplies exactly such a policy in Sections 12.4-12.5: back-pressure control (called max-weight when the network is single-hop), a rule that has become the default choice in the switching and wireless-scheduling literature because it requires no advance knowledge of arrival rates and achieves the largest possible stability region. This mission formalizes back-pressure control for the discrete-time packet network model and proves it maximally stable — the chapter's own counterpart to mission VIII's continuous-time Theorem 9.12.

Setting

At the start of each timeslot, having observed the current buffer contents $z$, the max-weight/ back-pressure (MW/BP) policy solves $\max_{s\in S(z)} z\cdot Rs$ (Eq. 12.41), where $S(z) := \{s\in S : Bs\le z\}$ restricts the schedule set $S$ (mission XII) to what is actually available given $z$. Equivalently (Eq. 12.42-12.43), writing $w_j(z) := z_{u(j)} - z_{d(j)}$ (with $z_0 := 0$) for the "weight" of activity $j$, the policy maximizes $\sum_j w_j(z) s_j$ — the schedule that clears the most "backlog pressure" per timeslot. The policy is deterministic (no randomization variable is needed, since (12.41) is a genuine optimization problem, not one that inherently requires randomized tie-breaking).

Formalization targets

Goal: Theorem 12.16 — aperiodicity, irreducibility, and conditional positive recurrence

Consider a packet network satisfying (12.1) and Assumption 12.1, operating under back-pressure control. The DTMC $Z$ is aperiodic and irreducible unconditionally. If the stability condition (12.26) — the existence of $\hat s\in\langle S\rangle$ with $\lambda < R\hat s$ — is additionally satisfied, $Z$ is positive recurrent. This is the mission's headline result, and it packages both a structural fact (irreducibility/aperiodicity, needed regardless of load) and a conditional stability fact (positive recurrence, needed only under (12.26)) in one theorem.

Supporting milestones

Lemma 12.17 shows the optimization problem (12.41) always has a nonzero solution whenever the network is nonempty — the fact that back-pressure never "idles unnecessarily," which Lemma 12.18 uses to show every state can reach the empty state (hence irreducibility) and that state $0$ has period $1$ (hence aperiodicity, via (12.1)'s own assumption that no external arrivals is possible with positive probability). Lemma 12.20 is the back-pressure fluid equation (12.44), the discrete-time analog of mission VIII's Theorem 9.8: at each regular point, the fluid-scaled buffer content dotted with the fluid departure rate equals the maximum of that same dot product over the convex hull of the schedule set. Lemma 12.21 shows this fluid model is stable whenever (12.26) holds, via an explicit quadratic Lyapunov function.

Significance

The result itself. Theorem 12.16 shows that back-pressure control — a rule requiring no knowledge of arrival rates, computed fresh from the current buffer contents every timeslot — is maximally stable: combined with Theorem 12.8 and Proposition 12.9 (mission XII), it stabilizes every packet network arrival-rate vector that any policy could possibly stabilize. This is the discrete-time, slotted analog of mission VIII's Theorem 9.12, and the two proofs share their essential Lyapunov argument (the book's own text calls Lemma 12.21's proof "almost identical to that of Theorem 9.12"), even though the two chapters' back-pressure equations are built from different underlying objects — mission VIII's continuous-time allocation polytope versus this chapter's convex hull of a discrete schedule set.

Formalizing it. A live prior-art check (GET /theorems?q=back-pressure, q=max-weight) finds no relevant hits, matching mission VIII's own finding for the continuous-time version. This mission formalizes the back-pressure optimization problem and policy, the back-pressure fluid equation, and its stability from scratch, restating (per this series' convention for concurrently-drafted chunks sharing a sub-namespace) the minimal apparatus needed from mission XII: the packet network model, Assumption 12.1, the raw processes, fluid limit paths, the fluid equations (12.31)-(12.36), and the ambient-chain machinery, plus a new Aperiodic predicate this chunk needs that mission XII's own results do not.

Difficulty

The obvious approach to Lemma 12.20's back-pressure fluid equation — directly differentiate the discrete system equation — obscures the actual argument, which reduces a maximum over the (potentially non-polytope) discrete schedule set $S$ to a maximum over its convex hull $\langle S\rangle$ (justified because a linear functional on a compact polytope is maximized at an extreme point) and then shows that every schedule with a strictly smaller objective value than the maximizer contributes zero derivative to the usage-counting process $\hat T_s$ — the discrete-time analog of mission VIII's Lemmas 9.10-9.11. A second difficulty is connecting the maximum-over-a- finite-set at the fluid-scaled level to Definition 12.3's discrete, arrival-indexed back-pressure policy: this mission's own Lemma 12.20 item states the fluid-relevant consequence of the discrete policy as a documented hypothesis (mirroring mission XI's own scope decision for its structurally identical WWTA fluid equation) rather than re-deriving it from the raw stochastic recursion, which would require reconstructing infrastructure outside this chapter's own numbered results.

Formalization scope

The packet network model, Assumption 12.1, the raw processes, and the fluid-limit apparatus are restated verbatim from mission XII (12-packet-networks-model), since concurrently-drafted chunks sharing a sub-namespace do not import one another's Lean files. IsBPOptimal is phrased via domination over the feasible-at-$z$ schedule set, not sSup/argmax, per this series' junk-value-avoidance convention; the same convention is applied to the maximum over $\langle S\rangle$ in the back-pressure fluid equation. The formalization does not admit a trivializing reading: irreducibility and aperiodicity are the same non-vacuous renewal-theoretic notions used throughout this series (Aperiodic requires a genuine positive self-transition probability, not a vacuously-true condition), and the goal's positive-recurrence conjunct is genuinely conditional on (12.26), stated with the correct strict inequality rather than weakened to $\le$. Contributions completing the five by sorry proofs are welcome, particularly Lemma 12.17's hop-count induction and Lemma 12.20's extreme-point/zero-derivative argument (mirroring mission VIII's own Lemmas 9.10-9.11, adapted to discrete time).

Selected references

• J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org

Motivation

Internet routers, wireless base stations, and data-switch fabrics all face the same recurring decision: in each discrete time slot, which of many possible transfer operations should be executed, given the packets currently queued and the physical constraints (link capacities, interference between simultaneous transmissions) on what can be done at once? J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) devotes Chapter 12 to exactly this question, under the name packet networks. Unlike every prior chapter of the book, which works in continuous time with Poisson-driven Markov chains, this chapter builds its stability theory from scratch for a discrete-time, slotted model — the natural setting for a system that makes one scheduling decision per clock tick. This mission formalizes the chapter's foundational layer: the model itself, its notion of a feasible schedule and control policy, subcriticality, and the discrete-time fluid-limit machinery that Chapters 13 and 14 (back-pressure control and random proportional scheduling, respectively) build their own stability proofs on top of.

Setting

A packet network (Section 12.1) has I packet classes and J activities (service types); activity $j$ transfers a packet from its input class $u(j)$ to its output class $d(j)$ (or removes it from the network if $d(j)=0$), giving an $I\times J$ input-output matrix $R$. A processing plan for class $i$ (Definition 12.2) is a chain of activities from $i$ to exit; Assumption 12.1 requires every class to have one and forbids cycles. In each timeslot the system manager chooses a schedule $s\in\mathbb Z^J_+$ from a feasible set $S$ satisfying the packet-availability constraint $Bs\le z$ (the current buffer contents); $S$ is typically built (Section 12.2) from a link usage matrix $A$ and a set $C$ of feasible link configurations via $S_c := \{s : As\le c\}$, $S := \bigcup_{c\in C} S_c$. A Markovian control policy (Definition 12.3, Eq. 12.8) chooses $s(\tau) = f(Z(\tau-1), U(\tau))$ from the current buffer contents and an independent randomization variable; it is admissible if it never overdraws a buffer, and stable if the resulting discrete-time Markov chain $Z$ is irreducible and positive recurrent.

Formalization targets

Goal: Theorem 12.10 — fluid limit stability implies positive recurrence

If the DTMC $Z$ under a Markovian policy $f$ is irreducible and its fluid limit is stable (Definitions 12.14-12.15), then $Z$ is positive recurrent. This is the chapter's own version of Theorem 6.2 (mission III) — the technical fulcrum that lets a deterministic fluid-model stability argument certify the stability of the original discrete, stochastic system — restated for a genuinely different probabilistic model, since Theorem 6.2 was built under continuous-time, Poisson-arrival hypotheses this chapter does not share.

Supporting milestones

Propositions 12.6-12.7 characterize the convex hulls $\langle S_c\rangle$ and $\langle S\rangle$ as explicit polytopes cut out by the link usage matrix — the combinatorial core that Theorem 12.8 (stability implies subcriticality, this chapter's analog of Theorem 5.2) and Proposition 12.9 (a necessary condition for subcriticality) both build on. Lemma 12.11 records that the schedule-usage counting process is Lipschitz; Lemma 12.12 is the functional strong law of large numbers the external arrival process satisfies; and Theorem 12.13 combines them to establish existence of discrete-time fluid limits satisfying the chapter's own fluid equations (12.31)-(12.36) — the chapter's analog of Theorem 6.5.

Significance

The result itself. Theorem 12.10 is what makes the rest of Chapter 12 (and Chapters 13-14) tractable: rather than analyzing an infinite-state discrete-time Markov chain's positive recurrence directly — a notoriously hard problem in general — it suffices to exhibit a deterministic fluid model and show every solution of that fluid model empties in finite time. This is the same strategy Chapter 6 established for the book's continuous-time model, but Chapter 12 cannot simply invoke that earlier theorem: the packet network model uses discrete time slots rather than a continuous clock, and its probabilistic structure (arbitrary i.i.d. arrival increments rather than Poisson arrivals) is different enough that the fluid-limit compactness argument has to be redone, even though — as the book's own text notes — "the proof mimics that of Theorem 6.2."

Formalizing it. A live prior-art check (GET /theorems?q=packet+network, q=discrete-time+Markov+chain, q=slotted+time) finds no relevant hits. A dedicated further check for MarkovMixing, a different mission's own corpus offering a PositiveRecurrent predicate for Markov chains, found a representationally distinct formalization (a row-function transition kernel rather than this series' own PMF-based jump-chain convention); this mission restates positive recurrence and irreducibility locally instead, consistent with every mission in this series since mission I. Everything else — the packet network model, schedules and configurations, the subcritical region, Markovian policies, and the discrete-time fluid-limit apparatus — is formalized from scratch.

Difficulty

The most consequential decision in this mission is representational, not mathematical: Theorem 12.10's fluid-limit-stability hypothesis quantifies over all fluid limit paths, which are themselves scaling limits of a genuinely stochastic discrete-time process — reconstructing that process from Chapter 2/4's own primitive stochastic elements (arrival processes, phase-type service mechanics) would require rebuilding infrastructure this chapter's own numbered results do not supply. Following mission III's own precedent for its structurally identical Theorem 6.5, this mission instead takes the raw, per-initial-state schedule-usage and arrival processes as given data satisfying only the recap properties the chapter's own proofs actually cite (Eq. 12.9-12.10's system equation, monotonicity), and fixes a single sample point together with an explicit SLLN hypothesis rather than a bare "for almost all $\omega$" quantifier — matching the book's own statement of Theorem 12.13, which itself begins "Fix an $\omega\in\Omega_1$" rather than quantifying almost everywhere within the theorem itself. A second difficulty is genuinely combinatorial: Proposition 12.6's proof constructs an explicit product-form probability distribution over schedules (one independent randomization per link) whose mean recovers an arbitrary point of the polytope $\{Ax\le c\}$ — a real combinatorial argument, not a formal consequence of the convex-hull operator's definition.

Formalization scope

Classes, activities, links, schedules and configurations are represented via Fin-indexed types throughout; S and C are taken as Finsets (every worked example in the book has finitely many configurations, and the link usage matrix's single-1-per-column structure bounds each schedule component, so this is a genuine, if implicit, standing feature of the model rather than an added restriction). IsScheduleSetAt/IsScheduleSet characterize $S_c$/$S$ via an ↔ against the underlying capacity constraint, rather than constructing them, matching how set-valued model data is handled elsewhere in this series. The formalization does not admit a trivializing reading: the ambient chain's positive recurrence (PositiveRecurrent) is the same non-vacuous renewal-theoretic notion used throughout this series, PacketFluidLimitStable quantifies over every genuine fluid limit path (not a hand-picked one), and subcriticalRegion is stated with a strict inequality ($A\hat s < \hat c$) exactly as Eq. 12.24 requires, not weakened to $\le$. Contributions completing the eight by sorry proofs are welcome, particularly Proposition 12.6's explicit randomized- schedule construction and Theorem 12.13's compactness argument (mirroring mission III's own Theorem 6.5 proof, adapted to discrete time).

Selected references

• J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org

Motivation

Large-scale data-intensive computing splits a job into many small "map tasks," each of which must be assigned to one of thousands of general-purpose servers in a data center — the MapReduce paradigm introduced by Dean and Ghemawat (2008). Because a task's input data is stored on only a few of those servers, routing it to a server that already holds its data ("data locality") avoids costly network transfer, so a task-allocation policy that ignores locality can leave a server farm badly underutilized even when raw compute capacity is ample. J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) devotes Chapter 11 to a rigorous stability analysis of Xie, Yekkehkhany, and Lu's (2016) workload-weighted task allocation (WWTA) policy, which routes each arriving task to whichever eligible server currently carries the least (locality-weighted) backlog. This mission formalizes the chapter's central result: WWTA is not merely a reasonable heuristic but is maximally stable — it keeps the system stable whenever any routing policy could.

Setting

A task allocation model (Sections 11.2-11.3) has L task categories and K servers; a class is a pair $(\ell,k)$, a category routed to a specific server, with mean service time $m_{\ell k} > 0$ (Eq. 11.3) reflecting both data locality (local, rack-local, or remote access) and the server's relative processing speed. Tasks in category $\ell$ arrive according to a Markovian arrival process (MArP) with long-run average rate $\nu_\ell$ — a substantially more general model than the independent Poisson streams used elsewhere in the book, needed to capture batches of map tasks generated from a single job. Routing is by immediate commitment: each task is assigned to a server the instant it arrives, based only on its category and the current backlog, with no possibility of later reassignment. The workload of server $k$ at time $t$ (Eq. 11.7) is $W_k(t) := \sum_\ell m_{\ell k} Z_{\ell k}(t)$, the locality-weighted total backlog assigned to it. Workload-weighted task allocation (WWTA), Definition 11.3, routes an arriving category-$\ell$ task to any server achieving $\operatorname{argmin}_{k} m_{\ell k} W_k(t-)$ (Eq. 11.8, ties broken arbitrarily), where $W(t-)$ is the workload vector's left limit at the arrival instant.

Formalization targets

Goal: Theorem 11.6 — WWTA fluid model stability under the load condition

If there exists $\lambda = (\lambda_{\ell k}) \ge 0$ with $\sum_k \lambda_{\ell k} = \nu_\ell$ for each category (Eq. 11.4) and $\sum_\ell m_{\ell k}\lambda_{\ell k} < 1$ for each server (Eq. 11.5), then the WWTA fluid model — the deterministic fluid equations (11.11)-(11.16) that arise as scaling limits of the stochastic model under WWTA control — is stable: every solution reaches the zero state in finite time proportional to its initial size.

Supporting milestones

Lemma 11.2 shows this load condition is equivalent to subcriticality of the task allocation model, in the sense of Section 5.2's general definition, reducing an existence statement over the model's full $G,R,A,b$ apparatus to a simple linear-feasibility check. Theorem 11.4 establishes that fluid limits of the stochastic model exist and satisfy the general fluid equations (11.11)-(11.15) under any immediate-commitment routing policy, and the WWTA-specific equation (11.16) under WWTA — the chapter's version of the fluid-limit machinery Chapter 6 builds for the book's standard model, adapted to Markovian (not Poisson) arrivals and immediate-commitment routing. Theorem 11.5 is the corresponding replacement for Theorem 6.2 (mission III): fluid limit stability implies stability of the original stochastic model. Corollary 11.7 is the mission's other headline consequence: WWTA is maximally stable, stabilizing the model whenever any simply structured routing policy could.

Significance

The result itself. A task-allocation heuristic motivated purely by an intuitive "balance-the-weighted-backlog" idea turns out to have the strongest stability guarantee a policy can have: it never needs re-tuning as arrival rates change, and no alternative policy can stabilize a regime WWTA cannot. Xie, Yekkehkhany, and Lu (2016) proved this fact (calling it "throughput optimality") by direct analysis of the discrete stochastic model using a quadratic Lyapunov function; Dai and Harrison's contribution is to prove the same conclusion via the fluid-model route, and — because the task allocation model violates two of the standing assumptions under which Chapter 6's general theory was built (MArP rather than Poisson arrivals, immediate-commitment routing rather than the book's default relaxed control) — to show that the general fluid-stability machinery survives those changes with only minor modification.

Formalizing it. A live prior-art check (GET /theorems?q=task+allocation, q=server%20farm, q=data%20locality) finds no relevant hits on the platform (workload returns one unrelated text-corpus-statistics result). This mission formalizes the task allocation model, WWTA, the fluid equations, and the stochastic-to-fluid bridge from scratch, reusing Mathlib's general real-analysis, measure-theory, and PMF-based Markov-chain substrate, and restating (per this series' convention for concurrently-drafted chunks) the minimal apparatus needed from missions I and III.

Difficulty

The obvious approach to Definition 11.3 — pick a canonical minimizer of $m_{\ell\cdot}W_\cdot(t-)$, e.g. via Finset.min' — silently narrows WWTA to a single tie-breaking rule, when (11.8) explicitly allows any minimizer; this mission instead states WWTA as domination over every server, accepting every admissible tie-break. A second difficulty is genuinely structural: Theorem 11.4's "for almost all $\omega$" existence claim is a measure-theoretic statement about a probability space that mission III's own analogous Theorem 6.5 sidesteps by fixing a sample point $\omega$ together with the specific SLLN limit that argument needs — this mission follows the same precedent rather than introducing a bare "almost everywhere" quantifier, which does not by itself supply the type information Lean's typeclass resolution needs to identify an ambient measure. A third is connecting Definition 11.3's discrete, arrival-indexed routing rule to the raw (pre-limit) process family that Theorem 11.4's existence argument operates on: the book's own proof derives the needed consequence (Eq. 11.22, "server $k$ receives no more category-$\ell$ tasks in a neighbourhood of $t$") via an explicit $\varepsilon$-$\delta$ argument from the discrete policy, which this mission takes as a documented hypothesis rather than re-deriving, since building the arrival-sequence-to-counting-process reconstruction is genuinely Chapter 2/4 machinery outside this chapter's own numbered results.

Formalization scope

Classes are represented as pairs (ℓ,k) : Fin L × Fin K rather than flattened to a single index, avoiding an arbitrary choice of encoding. TaskAllocationProcessFamily, FluidLimitPath, and TaskAllocationFluidLimitStable restate mission III's SPNProcessFamily/FluidLimitPath/ FluidLimitStable apparatus, adapted to this model's (E,D,Z) triple and doubly-indexed classes; PositiveRecurrent restates mission I's renewal-theoretic positive-recurrence notion. The formalization does not admit a trivializing reading: IsWWTA accepts every admissible tie-break rather than a single canonical one, SatisfiesWWTAFluidEquation requires Nonempty (Fin K) so its min_{k} is a genuine minimum rather than a junk value at zero servers, and the goal's conclusion WWTAFluidStable is the same non-vacuous fluid-model-stability predicate (Definition 6.3, specialized) used throughout this series. Contributions completing the five by sorry proofs are welcome, particularly Theorem 11.4's fluid-limit compactness argument (mirroring mission III's own Theorem 6.5 proof) and Theorem 11.6's explicit quadratic-Lyapunov-function argument.

Selected references

• J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
• J. Dean and S. Ghemawat, "MapReduce: simplified data processing on large clusters," Communications of the ACM 51 (2008), 107–113.
• Q. Xie, A. Yekkehkhany, and Y. Lu, "Scheduling with multi-level data locality: throughput and heavy-traffic optimality," IEEE INFOCOM 2016.

Motivation

An inventory manager who stocks many products, faces random demand and pays ordering, holding and shortage costs must decide each period how much of each product to order. In general the optimal decision depends on the whole state and is found by solving a dynamic programme, which becomes impractical as the number of products grows. Myopic (or base stock) policies avoid this: in each period they aim at a target level computed from that period's data alone. Knowing when such a policy is optimal over an infinite horizon tells a practitioner when the multi-period problem decouples into a sequence of one-period problems.

Arthur F. Veinott, Jr. gave such conditions in Optimal Policy for a Multi-Product, Dynamic, Nonstationary Inventory Problem (Management Science 12(3):206–222, 1965). Earlier results of this type were for a single product (the paper cites Karlin, Management Science 6(3), 1960, Bellman–Glicksberg–Gross, Management Science 2(1), 1955, Iglehart–Karlin 1962, and the Arrow–Karlin–Scarf volume of 1958); Veinott's model allows several products, several demand classes, costs and demand distributions that change over time, general ordering constraints and general stock dynamics (backlogging, lost sales, and mixtures), and it does not use the functional equation of dynamic programming. Instead, the proofs analyse the inventory process directly.

Setting

There are $n$ products and $m$ demand classes. Vectors are compared componentwise: $u \le v$ means $u_j \le v_j$ for all $j$. In period $i = 1, 2, \dots$ the manager observes the inventory vector $x_i \in X_i \subseteq \mathbb{R}^n$ (negative coordinates are backlogs) and orders up to a vector $y_i \in Y_i \subseteq \mathbb{R}^n$, subject to $y_i \ge q_i(x_i)$, where $q_i$ is a vector of extended-real functions (for instance $q_i(x) = x$ forbids disposal). A random demand vector $D_i$ with law $\Phi_i$ and values in a Borel set $\mathfrak{D}_i \subseteq \mathbb{R}^m$ then occurs, and the next inventory vector is $x_{i+1} = s_i(y_i, D_i) \in X_{i+1}$. The demands $D_1, D_2, \dots$ are independent. Ordering $y_i - x_i$ costs $c_i \cdot (y_i - x_i)$, the holding and shortage cost is $g_i(y_i, D_i)$, and $\alpha_i \ge 0$ is the discount factor of period $i$.

Regrouping the ordering costs gives the one-period cost

$$W_i(y,t) = c_i\, y + g_i(y,t) - \alpha_i\, c_{i+1}\, s_i(y,t), \qquad G_i(y) = \int_{\mathfrak{D}_i} W_i(y,t)\, d\Phi_i(t),$$

with discount weights $\beta_1 = 1$ and $\beta_i = \alpha_1 \cdots \alpha_{i-1}$. The integrals are assumed finite, and $G_i \ge \gamma_i$ with $\sum_i |\beta_i \gamma_i| < \infty$. An ordering policy $\bar Y$ chooses $y_i$ as a Borel function of the past; it is feasible if $y_i \in Y_i$ and $y_i \ge q_i(x_i)$ for every possible history. Its cost is

$$f(x_1 \mid \bar Y) = \sum_{i=1}^\infty \beta_i\, E\, G_i(y_i) \in (-\infty, +\infty],$$

and a feasible policy of least cost is optimal.

Let $\bar y_i$ minimize $G_i$ over $Y_i$. When $\bar y_i$ is not attainable from $x$, the minimal feasible level $w_i(x)$ is the least element of $Y_i \cap \{y : y \ge q_i(x),\ y \ge \bar y_i\}$. The base stock ordering policy orders up to $\bar y_i$ if $q_i(x_i) \le \bar y_i$ and up to $w_i(x_i)$ otherwise.

Formalization targets

The hypotheses are: (3a) $\bar y_i \in Y_i$ minimizes $G_i$ over $Y_i$; (3b) $q_{i+1}(s_i(\bar y_i, t)) \le \bar y_{i+1}$ for $t \in \mathfrak{D}_i$; (3c) $Y_i$ is closed and linearly ordered by $\le$; (3d) $G_i$ and $s_i(\cdot, t)$ are nondecreasing on $\{y \in Y_i : y \ge \bar y_i\}$, and $q_i$ is nondecreasing where $q_i(x) \not\le \bar y_i$.

Goal: Theorem 3.2

Under (3a)–(3d), the base stock ordering policy

$$\bar Y_i^*(H_i^*) = \begin{cases} \bar y_i, & q_i(x_i^*) \le \bar y_i, \\ w_i(x_i^*), & q_i(x_i^*) \not\le \bar y_i \end{cases}$$

is feasible and optimal: $f(x_1 \mid \bar Y^*) \le f(x_1 \mid \bar Y)$ for every feasible $\bar Y$.

Milestones

1. A nonempty, closed, linearly ordered, bounded-below subset of $\mathbb{R}^n$ has a least element (p. 212); hence $w_i(x)$ exists and equals $\bar y_i$ when $q_i(x) \le \bar y_i$.
2. $w_i$ is nondecreasing where $q_i(x) \not\le \bar y_i$ (p. 214).
3. Once $q_k(x_k^*) \le \bar y_k$, the base stock policy orders up to $\bar y_i$ for all $i \ge k$.
4. Theorem 3.1: under (3a) and (3b), if $q_1(x_1) \le \bar y_1$, ordering up to $\bar y_i$ in every period is optimal, with cost $\sum_i \beta_i G_i(\bar y_i)$.
5. The coupling (3.1): before the first period $T$ with $q_T(x_T^*) \le \bar y_T$,

$$\bar y_i < y_i^* = w_i(x_i^*) \le w_i(x_i) \le y_i.$$

6. Pathwise dominance: $G_i(y_i^*) \le G_i(y_i)$ for every $i$ and every possible demand path.
7. The reduction behind (2.3): $E\, W_i(y_i, D_i) = E\, G_i(y_i)$, by independence of $y_i$ and $D_i$.

Significance

The theorem shows that under (3a)–(3d) the infinite-horizon, nonstationary, multi-product problem is solved by one-period optimization: compute $\bar y_i$ from $G_i$ alone, and when it is unreachable order the least feasible amount. No value function is computed. It covers backlogging and lost sales, products stocked in fixed proportions, and time-varying cost and demand data, and Theorem 3.1 alone settles the frequent case in which the initial stock is small.

The result is proved in the paper. To our knowledge it has not been machine-checked: this mission produces a Lean model of a general stochastic, infinite-horizon, multi-product inventory problem (feasible policies, the expected discounted cost with $+\infty$ allowed), and a checked proof that a myopic policy is optimal in it. The model and its cost are reusable for other base stock and myopic optimality results.

Difficulty

The obvious argument would compare the base stock policy with an arbitrary policy period by period using (3a) alone. That fails once $\bar y_i$ is unreachable. The base stock policy then holds less stock than $\bar y_i$ would require, and it has to be shown that no other policy can reach a lower-cost level later. The proof couples the two trajectories along every demand path, using the monotonicity in (3d) and the linear order of $Y_i$ from (3c), until the base stock level becomes attainable, after which (3b) keeps it attainable. The formal difficulties are:

• the existence and monotonicity of the least element $w_i(x)$ in a closed chain of $\mathbb{R}^n$;
• the measurability of the base stock policy, which is part of its feasibility;
• the passage from pathwise dominance to the expected discounted cost when the cost may be $+\infty$.

Formalization scope

Periods are 0-based in Lean (Lean period $k$ is the paper's period $k+1$). Vectors are Fin n → ℝ with the product order; $q_i$ takes values in Fin n → EReal. Policies are functions of the past demands. The paper shows on p. 219 that this loses no generality once $x_1$ is fixed. The cost is excess, a $[0,\infty]$-valued series of lower Lebesgue integrals of $\beta_i(G_i(y_i) - \gamma_i)$, plus the finite real $\sum_i \beta_i\gamma_i$, taken in EReal. A divergent series therefore gives $+\infty$ and never the junk value $0$. "Minimal element" is IsLeast. The $G_i$ in the statements is the integral of $W_i$ built from the data, and the policy in the goal is constructed from $\bar y$, $q$, $w$ and $s$. Neither is an arbitrary object satisfying the conclusion. The goal concludes optimality, which includes feasibility: a pathwise or conditional statement would not be the theorem.

Hypotheses added relative to the page, each used by the paper without being stated:

• Order feasibility: from every $x \in X_i$ some $y \in Y_i$ with $y \ge q_i(x)$ exists (p. 212 asserts the set defining $w_i(x)$ has a minimal element, which needs it nonempty). The least-element lemma likewise assumes $A$ nonempty.
• (3d)'s $q$-clause in one-sided form: if $x \le x'$ in $X_i$ and $q_i(x) \not\le \bar y_i$, then $q_i(x) \le q_i(x')$. This is the form the proof on p. 214 uses; monotonicity on the region $\{q_i(x) \not\le \bar y_i\}$ alone admits a counterexample to Theorem 3.2.
• Integrability of $W_i(y_i, D_i)$ in the milestone $E W_i = E G_i$.
• Borel measurability of $q_i$ on all of $\mathbb{R}^n$ rather than on $X_i$ (a harmless strengthening).

A complete development needs: least elements of closed chains in $\mathbb{R}^n$; measurability of the least-element selection $x \mapsto w_i(x)$; a Fubini/independence argument for $E\,W_i(y_i,D_i)=E\,G_i(y_i)$; and monotone summation of lower integrals. The least-element lemma and the cost encoding are reusable beyond this mission. Proofs of any milestone are welcome, as are alternative arguments.

Selected references

• A. F. Veinott, Jr., Optimal Policy for a Multi-Product, Dynamic, Nonstationary Inventory Problem, Management Science 12(3):206–222, 1965. https://doi.org/10.1287/mnsc.12.3.206
• R. Bellman, I. Glicksberg, O. Gross, On the Optimal Inventory Equation, Management Science 2(1):83–104, 1955. https://doi.org/10.1287/mnsc.2.1.83
• S. Karlin, Dynamic Inventory Policy with Varying Stochastic Demands, Management Science 6(3):231–258, 1960. https://doi.org/10.1287/mnsc.6.3.231

Motivation

Mission IX (09-proportional-fairness-core) formalized proportional fairness (PF) as a control policy and proved the technical core of its stability theory: under a load condition, the PF fluid model is stable (Theorem 10.5), via an entropy Lyapunov function that is genuinely not Lipschitz continuous — a departure from every other stability argument in J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org). A stability theorem for one fixed arrival-rate vector is, on its own, a narrower claim than practitioners actually want: real systems see load that changes over time, and a control policy worth adopting should not need re-tuning every time the mix of traffic shifts. This mission completes Theorem 10.5's proof and turns it into exactly that stronger guarantee — proportional fairness is maximally stable: stable throughout the entire region where any policy could be stable, without knowing the arrival rates in advance — and specializes the result to two concrete network families, bandwidth-sharing networks and queueing networks under head-of-line proportional processor sharing (HLPPS), that were already familiar from earlier in the book under different control policies.

Setting

Fix a unitary network operating under PF control with mean service times $m > 0$, routing matrix $P$, a partition of job classes into demand groups $\{\mathcal I(\ell), \ell \in \mathcal L\}$, and a reduced allocation set $\tilde{\mathcal A} \subset \mathbb R^{\mathcal L}_+$ (all restated from mission IX, Definition 10.3). The entropy Lyapunov function $\varphi(t) := \sum_i Z_i(t)\log(\dot D_i(t)/\alpha_i)$ (Eq. 10.38, mission IX) admits an alternative decomposition $\varphi = \sum_\ell \varphi_\ell$ in terms of the within-group entropy term

$$f(t) := \sum_{\ell\in\mathcal L}\sum_{i\in\mathcal I(\ell)} Z_i(t)\log\!\left(\frac{Z_i(t)}{Y_\ell(t)}\right), \qquad Y(t) := GZ(t) \quad \text{(Eq. 10.50)},$$

with the convention that the term for class $i$ is $0$ when $Z_i(t)=0$. Here $D^+$/$D^-$ denote the upper-right/upper-left Dini derivatives (Appendix A.4, Eqs. A.9-A.10): at a point where the ordinary derivative may not exist, these one-sided $\limsup$s still let a Lyapunov-drift argument go through. A control policy is maximally stable (Section 5.7) for a network if its implementation does not depend on the arrival-rate vector $\lambda$ and it is stable for every $\lambda$ in the network's stability region $\Lambda^*$ — the largest region any policy could possibly stabilize.

Formalization targets

Goal: Corollary 10.16 — maximal stability of PF control for a unitary network

$$\text{IsMaximallyStable}\Big(\lambda \mapsto \text{PFFluidStable}(\lambda, m, P, \mathrm{grp}, \tilde{\mathcal A})\Big)$$

for the single PF policy value (its implementation never depends on $\lambda$). This is the applied payoff of Theorem 10.5 (mission IX): combined with Theorem 6.2 (mission III, fluid stability implies SPN stability) and Corollary 5.6 (mission II, a $\lambda$-independent policy stable throughout the subcritical region is automatically maximally stable), it upgrades a single-$\lambda$ stability statement to the strongest form the book's own framework can express.

Supporting milestones

Lemmas 10.11-10.15 supply the remaining technical content of Theorem 10.5's proof that mission IX's own milestones left open: Lemma 10.11 is the uniform negative-drift bound $\sum_i \dot Z_i(t)\log(\dot D_i(t)/\alpha_i) \le -\varepsilon$ at every regular point with $Z(t) \ne 0$; Lemmas 10.12-10.14 establish continuity and two successively sharper Dini-derivative bounds on the within-group entropy term $f$; Lemma 10.15 shows that positivity of the departure rate $\dot D_i(t)$ propagates from occupied classes to every class. Corollaries 10.17 and 10.18 specialize the goal to bandwidth-sharing networks and to HLPPS-controlled queueing networks, respectively.

Significance

The result itself. A stability theorem tied to one fixed $\lambda$ is of limited practical use: it would need to be re-verified every time the arrival-rate vector changes, which real traffic does constantly. Maximal stability removes that dependency entirely — a single policy, implemented without any knowledge of $\lambda$, is guaranteed stable throughout the full region any control could stabilize. Corollaries 10.17 and 10.18 make this concrete for two network families with independent histories in the literature: bandwidth-sharing networks (the original motivation for proportional fairness, Kelly 1997) and queueing networks under HLPPS, connecting PF's static, utility-theoretic motivation to a scheduling rule that predates it.

Formalizing it. A live prior-art check (GET /theorems?q=proportional%20fairness, q=maximal%20stability, q=bandwidth%20sharing) finds no relevant hits on the platform. This mission formalizes the remaining entropy-Lyapunov lemmas, the within-group entropy term, the BWS and HLPPS network models (restated locally, since no other drafted chunk covers Sections 4.5-4.6), and the maximal-stability predicate, reusing only Mathlib's general real-analysis substrate (Dini derivatives via Filter.limsup) and definitions restated from missions II, III, V, and IX under this series' restate-not-import convention for concurrently-drafted chunks.

Difficulty

The obvious approach to Corollary 10.16 — restate "maximally stable" with an explicit $\lambda$-dependent policy family and add "the policy doesn't actually depend on $\lambda$" as a side hypothesis — obscures the point: a policy that is definitionally independent of $\lambda$ is a stronger and cleaner claim than one that happens to satisfy an extra equation. This formalization instead instantiates the abstract policy type at Unit, so $\lambda$-independence holds by construction rather than as a hypothesis to verify, matching the book's own reading of Section 5.7's definition. A second difficulty is Lemma 10.13's Dini-derivative inequality (Eq. 10.56): the plain-text extraction of this display equation loses bracket and subscript structure that changes its meaning, so the exact grouping was confirmed against the PDF page directly and cross-checked against the book's own re-derivation of the same bracketed expression inside Lemma 10.14's proof. A third is Corollary 10.17's proof, which genuinely depends on three facts outside this chunk's own chapter portion (Proposition 4.4 and the Section 4.4 model translation, Proposition 5.1, Theorem 5.2); rather than silently assuming them or re-deriving their proofs from scratch, they are stated as explicit hypotheses of the milestone itself, so the item's actual content — deriving the two-sided conclusion — is exactly what remains to be proved.

Formalization scope

RestatedCore, RestatedFluidModel, and the maximal-stability predicate MaximalStability are restated verbatim (or, for MaximalStability, in shape) from missions IX, III/V, and II respectively, since concurrently-drafted chunks in this series do not import one another's Lean files even when they share a sub-namespace. New to this chunk: diniUpperLeft (Eq. A.10, needed alongside mission IX's diniUpperRight for Lemma 10.13's two-sided bound), the within-group entropy term withinGroupEntropy (Eq. 10.50, deliberately not named f, since the book's own f already denotes the unrelated PF optimization objective of Eq. 10.2 within the same chapter), and BWSNetworkData/QueueingNetworkDataHL/HLPPSFluidStable (Sections 4.5-4.6, restated locally since no drafted chunk's BRIEF.md covers them). The formalization does not admit a trivializing reading: IsMaximallyStable is instantiated with the genuine, non-vacuous predicate PFFluidStable/HLPPSFluidStable — the same predicate whose stability Theorem 10.5 (mission IX) already establishes on the load-condition region — never with a policy type or stability predicate engineered to make the maximal-stability claim vacuous. Contributions completing the eight by sorry proofs are welcome, particularly Lemma 10.13's Dini-derivative estimate (Section B.4's preliminary results) and Lemma 10.15's connectivity argument (Appendix B.9/B.17).

Selected references

• J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
• F. P. Kelly, "Charging and rate control for elastic traffic," European Transactions on Telecommunications 8 (1997), 33–37.
• F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, "Rate control for communication networks: shadow prices, proportional fairness and stability," Journal of the Operational Research Society 49 (1998), 237–252.

Motivation

Dynamic pricing under limited inventory is a core problem of revenue management: a seller holds $C_0$ units of a perishable product (airline seats, hotel rooms, seasonal goods) and must choose prices over a finite selling horizon while demand is random and responds to price. The exactly optimal policy solves a stochastic dynamic program whose state is the remaining inventory, and it changes the price after every sale. In practice sellers often prefer a simpler rule, fixed in advance: solve the deterministic version of the problem, in which random demand is replaced by its mean, and charge the resulting prices whatever happens.

Gallego and van Ryzin (Management Science 1994) showed that in continuous time with Poisson demand this fixed-price heuristic is asymptotically optimal, and bounded its relative loss by the coefficient of variation of demand. Bitran and Caldentey's survey (MSOM 2003, §3.2.1) extends this bound to a discrete-time, periodic-review model with general demand distributions, as Proposition 8, proved in the paper's Appendix. The proof combines three ingredients: a Lagrangian duality argument showing that the deterministic problem is an upper bound on the optimal expected revenue, a sample-path comparison of lost sales, and a distribution-free moment bound of Gallego (1992) on $E[(X-C)^+]$.

Setting

A single product is sold over $N \ge 1$ periods $n = 1, \dots, N$ starting from inventory $C_0$. In period $n$ the seller charges a price $p \ge 0$, and the demand $D_n(p)$ is a nonnegative random variable with finite mean $E[D_n(p)]$, whose law may depend on $n$ and $p$ arbitrarily. With inventory $C$ the seller sells $\min\{D, C\}$ units; unmet demand is lost.

The optimal expected revenue $V_1(C_0)$ is defined by the Bellman recursion $V_{N+1} \equiv 0$,

$$V_n(C) = \sup_{p \ge 0} E\big[p\min\{D_n(p), C\} + V_{n+1}\big(C - \min\{D_n(p), C\}\big)\big].$$

The deterministic problem (32)–(33) is

$$V_1^{\det}(C_0) = \sup_{p \in [0,\infty)^N} \sum_{n=1}^N p_n E[D_n(p_n)] \quad \text{subject to} \quad \sum_{n=1}^N E[D_n(p_n)] \le C_0,$$

and $p^{\det}$ denotes an optimal solution. The deterministic-price heuristic charges $p^{\det}_n$ in period $n$ regardless of sales. Its period demands $D_n(p_n^{\det})$ are independent, $\mathscr{D}_n^{\det} = \sum_{i=1}^n D_i(p_i^{\det})$ is the cumulative demand, and its expected revenue is

$$V_1(p^{\det}, C_0) = \sum_{n=1}^N p_n^{\det} E\Big[D_n(p_n^{\det}) - \big(D_n(p_n^{\det}) - (C_0 - \mathscr{D}_{n-1}^{\det})^+\big)^+\Big].$$

With $\sigma_n^2 = \operatorname{Var}(\mathscr{D}_n^{\det})$, eq. (34) defines

$$\eta_n^{\det}(C_0) = \frac{\sqrt{\sigma_n^2 + (C_0 - E[\mathscr{D}_n^{\det}])^2} - (C_0 - E[\mathscr{D}_n^{\det}])}{2},$$

and $\nu(C_0) = \sigma_N / E[\mathscr{D}_N^{\det}]$ is the coefficient of variation of the total demand.

Formalization targets

Goal: Proposition 8, eq. (35)

Assume that $p \mapsto p\,E[D_n(p)]$ is concave and $p \mapsto E[D_n(p)]$ is convex on $[0,\infty)$ for each $n$, that some price $p^\infty \ge 0$ has $\sum_n E[D_n(p^\infty)] < C_0$, and that $p^{\det}$ is optimal for (32)–(33). Then

$$1 \ge \frac{V_1(p^{\det}, C_0)}{V_1(C_0)} \ge \frac{1}{V_1^{\det}(C_0)} \sum_{n=1}^N p_n^{\det} E[D_n(p_n^{\det})]\left(1 - \frac{\eta_n^{\det}(C_0)}{E[D_n(p_n^{\det})]}\right) \ge 1 - \max_n \frac{\eta_n^{\det}(C_0)}{E[D_n(p_n^{\det})]}.$$

The constants are the paper's, and the goal is the full chain.

Milestones

1. Gallego's bound (Appendix, (*)): for square-integrable $X$, $E[(X - C)^+] \le \frac12\big(\sqrt{\operatorname{Var}X + (C - EX)^2} - (C - EX)\big)$, which is at most $\frac12\sqrt{\operatorname{Var} X}$ when $EX \le C$.
2. Eq. (24): in one period, $E[p\min\{D(p), C\}] \le p\min\{E[D(p)], C\}$, so $V(C) \le V^{\det}(C)$.
3. Proposition 6, eq. (26): in one period, $V(C, p^{\det})/V(C) \ge 1 - \nu^{\det}/2$.
4. $V_1^{\det}$ is concave in the capacity (first sentence of the Appendix proof).
5. Proposition 8, first assertion: $V_1(C_0) \le V_1^{\det}(C_0)$.
6. The Appendix display bounding $V_1(p^{\det}, C_0)$ below by $\sum_n p_n^{\det}E[D_n](1 - E[(\mathscr{D}_n^{\det} - C_0)^+]/E[D_n])$.
7. Eq. (36), after the goal: if $E[D_n(p)] = T_n\lambda(p)$, one constant price solves (32)–(33) and $1 \ge V_1(p^{\det},C_0)/V_1(C_0) \ge 1 - \nu(C_0)/2$.

Significance

The result gives a guarantee for a pricing policy that needs no inventory tracking: its relative loss is controlled by the first two moments of cumulative demand, with no distributional assumption beyond finite variance. When demand grows while its coefficient of variation shrinks, as for sums of independent period demands, the guarantee tends to one, which is the discrete-time form of asymptotic optimality of fixed prices. The upper bound $V_1 \le V_1^{\det}$ is used throughout revenue management as the benchmark for heuristics (fluid or deterministic LP bounds).

The paper's proof is complete in the Appendix, and the mathematics is not in question beyond minor typos. None of it is machine-checked. A formal development adds a checked Bellman model of periodic-review pricing with general demand laws, a checked fluid upper bound for it, and a checked form of Gallego's moment bound, all reusable for other revenue-management statements. A related platform theorem, RevenueManagement.deterministic_upper_bound (mission The Theory and Practice of Revenue Management III), states the deterministic bound for a Bernoulli-arrival model with at most one sale per period; the model here is different (arbitrary demand laws, continuous inventory), and neither statement implies the other.

Difficulty

The upper bound $V_1(C_0) \le V_1^{\det}(C_0)$ is the hard part. The natural induction replaces $V_2$ by $V_2^{\det}$ and applies Jensen's inequality, but $V_2^{\det}(C)$ is $-\infty$ at capacities where the remaining deterministic problem is infeasible, while $V_2(C)$ stays nonnegative. The induction therefore fails as stated when mean demand never vanishes. The paper's argument also passes through the Lagrangian dual of (32)–(33), and strong duality for that program on $[0,\infty)^N$ under the Slater point $p^\infty$ is not available in Mathlib in this form.

The lower bound is less deep but technical: it needs the product law of the period demands, linearity of expectation for truncated sums, and variances of partial sums. Gallego's inequality is elementary once the right quadratic bound on $(x - C)^+$ is found, but it is not in Mathlib.

Formalization scope

• Periods are Fin N, 0-based. Demand laws are μ n p : Measure ℝ, probability measures carried by $[0,\infty)$ with finite mean, required at every price (laws at negative prices never enter a statement).
• The Bellman value is computed in $[0,\infty]$ with the lower Lebesgue integral and ⨆ over $p \ge 0$, so no supremum takes a junk value; the paper's $V_n$ is valueToGo M (N - n + 1). Ratios use its real value.
• $V_1^{\det}$ is an EReal supremum over the feasible set ($-\infty$ if infeasible). In the goal $p^{\det}$ is an optimal solution, so $V_1^{\det}(C_0)$ is its objective value.
• The heuristic's demands are independent: the joint law is the product measure.
• Implicit hypotheses made explicit: "concave objective and convex feasible region" is read as concavity of $p\,E[D_n(p)]$ and convexity of $E[D_n(p)]$ on $[0,\infty)$, the latter making (33) convex for every capacity; existence of the optimal deterministic solution; finite variance and positive mean of each $D_n(p_n^{\det})$; and $V_1^{\det}(C_0) > 0$, without which the ratios are $0/0$.
• The typo $\mathscr{D}_n^{\det} := \sum_{i=1}^n D_n(p^{\det})$ on p. 221 is read as $\sum_{i=1}^n D_i(p_i^{\det})$.

Trivializing formalizations ruled out: $V_1$ is defined by the Bellman recursion, not as a supremum over an unspecified policy class or as a variable constrained by hypotheses; $\eta_n^{\det}$ is the expression (34), not a hypothesis-supplied bound on expected overflow; the goal does not assume strong duality or a Lagrange multiplier, since that is the proof's key step; independence is built into the joint law, not assumed as an inequality; variances are taken only under square integrability, since Mathlib's variance is $0$ for infinite variance.

Needed infrastructure: measurability of the Bellman integrand (monotonicity of $V_n$ in inventory), Jensen's inequality for $\min\{\cdot, C\}$ (ConcaveOn.le_map_integral), Lagrangian strong duality for a separable concave program with one convex constraint, marginals and variances of sums under Measure.pi, and Gallego's bound. The duality and moment results are reusable beyond this mission. Proofs of any milestone, and alternative proofs of the upper bound, are welcome.

Selected references

• G. R. Bitran, R. Caldentey, An Overview of Pricing Models for Revenue Management, Manufacturing & Service Operations Management 5(3):203–230, 2003. https://doi.org/10.1287/msom.5.3.203.16061
• G. Gallego, G. van Ryzin, Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons, Management Science 40(8):999–1020, 1994. https://doi.org/10.1287/mnsc.40.8.999
• G. Gallego, A Minmax Distribution Free Procedure for the (Q, R) Inventory Model, Operations Research Letters 11(1):55–60, 1992 (as cited in Bitran and Caldentey 2003).
• M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd ed., Wiley, 1993.