Optimization

Motivation

Many production and computing systems run jobs on parallel machines that also draw on a shared, limited resource: tools, workers, memory, power. Adding such a resource to a scheduling problem can change its complexity entirely. Błażewicz, Lenstra and Rinnooy Kan (DAM 1983) extended the three-field classification $\alpha\mid\beta\mid\gamma$ of Graham, Lawler, Lenstra and Rinnooy Kan by a resource field $res\lambda\sigma\rho$, and determined the complexity of every problem with unit-time jobs on identical or uniform machines under the makespan criterion. Their Fig. 2 separates the maximal polynomially solvable cases from the minimal NP-hard ones.

This mission formalizes the polynomial side. Two identical machines are easy under arbitrary resources (Theorem 1, due to Garey and Johnson, via maximum matching). Three identical machines with one resource are NP-hard in the strong sense (Theorem 4), and so are two uniform machines with unit resources (Theorem 3). What remains for uniform machines is settled by two algorithms: a sorting-and-shifting procedure for two uniform machines with one resource of arbitrary size (Theorem 5), and a bottleneck transportation problem for any number of uniform machines with one resource and 0–1 requirements (Theorem 6). The hardness results are the subject of the companion missions I and II.

Setting

There are $n$ jobs $J_1,\dots,J_n$ and $m$ machines $M_1,\dots,M_m$. Machine $M_i$ has speed $q_i>0$; every job has unit execution requirement, so it takes time $1/q_i$ on $M_i$. Identical machines ($P$) have $q_i=1$; uniform machines ($Q$) have arbitrary speeds. There are $l$ resources $R_h$ with positive integer sizes $s_h$, and job $J_j$ needs a nonnegative integer amount $r_{hj}$ of $R_h$ throughout its execution. The field $res\lambda\sigma\rho$ records restrictions: $\lambda$ bounds the number of resources, $\sigma$ their sizes, $\rho$ the requirements, a dot meaning "part of the input". So $res1{\cdot}{\cdot}$ is one resource with arbitrary size and requirements, and $res1{\cdot}1$ is one resource with requirements in $\{0,1\}$.

A schedule gives every job a machine $\mu(j)$ and a start time $S_j\ge0$; the job is executed during $[S_j,C_j)$ with $C_j=S_j+1/q_{\mu(j)}$. It is feasible if jobs on the same machine do not overlap and, at every time $t$, the jobs executed at $t$ use at most $s_h$ of each resource $R_h$. The makespan is $C_{\max}=\max_j C_j$. No precedence constraints occur in this mission.

Formalization targets

Goal: Theorem 5, correctness of the algorithm

For $Q2\mid res1{\cdot}{\cdot},\,p_j=1\mid C_{\max}$ with $q_1\ge q_2$: put all jobs on $M_1$ in order of nonincreasing $r_{1j}$, then repeatedly move the last job of $M_1$ to the earliest feasible time on $M_2$ after the jobs already there, as long as this strictly reduces $C_{\max}$. For every order with nonincreasing requirements, the resulting schedule $A$ is feasible and

$$C_{\max}(A)\le C_{\max}(\sigma)\quad\text{for every feasible schedule }\sigma.$$

Milestones for the goal

The paper's proof has two steps, both milestones. Call a schedule an (a)–(c) schedule when (a) $M_1$ runs its jobs back to back from time $0$ in nonincreasing $r_{1j}$, (b) $M_2$ runs its jobs in nondecreasing $r_{1k}$, and (c) every requirement on $M_1$ is at least every requirement on $M_2$.

1. The algorithm's schedule is feasible, is an (a)–(c) schedule, and is best among feasible (a)–(c) schedules.
2. Every feasible schedule can be transformed into a feasible (a)–(c) schedule with no larger $C_{\max}$.

Further results

• Theorem 1. For $P2\mid res{\cdot}{\cdot}{\cdot},\,p_j=1\mid C_{\max}$, with $G$ the graph joining two jobs when they can run together and $S$ a maximum matching of $G$, the optimal makespan is $n-|S|$.
• Theorem 6. For $Q\mid res1{\cdot}1,\,p_j=1\mid C_{\max}$ with the $s_1$ fastest machines listed first, the optimal makespan equals the optimal value of a bottleneck transportation problem that assigns jobs to slots (machine, position) with cost $k/q_i$, resource jobs only to the $s_1$ fastest machines.

Significance

Theorems 5 and 6 complete the classification of Fig. 2 for uniform machines: every special case of $Q\mid res{\cdot}{\cdot}{\cdot},\,p_j=1\mid C_{\max}$ not covered by the hardness theorems has a polynomial algorithm. Theorem 1 is the classical reduction of two-machine resource scheduling to maximum matching, the model case for later work on scheduling with conflict graphs.

The paper proves these results briefly: "clearly" for the first half of Theorem 5, "obviously" for Theorem 1, and a one-paragraph model for Theorem 6. The exchange argument of Theorem 5 is presented "in an informal way" through five steps that pass through fractional, preempted jobs. A machine-checked proof makes these arguments exact on a model with real start times. No formalization of these results is known, and the platform had no statement about resource-constrained scheduling on uniform machines before this mission.

Difficulty

With $q_1\ne q_2$ the job boundaries on the two machines are misaligned: a job on $M_2$ overlaps parts of several jobs on $M_1$, so the resource check cannot be done slot by slot, and discrete reasoning on integer time grids does not apply. The exchange argument of Theorem 5 must control the resource usage at every real time while jobs are moved between machines and reordered, and it has to end with a nonpreemptive schedule even though the paper's intermediate steps split jobs. For Theorem 1, the hard direction is the lower bound: a feasible schedule with arbitrary real start times must be converted into a matching, which is a statement about how unit jobs on two machines can overlap. For Theorem 6, one must show that restricting resource jobs to the fastest machines and to back-to-back positions loses nothing.

Formalization scope

• Model. Jobs, machines and resources are Fin n, Fin m, Fin l (0-based). Speeds are positive reals, sizes positive naturals, requirements naturals. Start times are nonnegative reals, execution intervals are half-open, and the resource constraint is checked at every real time. $C_{\max}=0$ for $n=0$. The model carries a precedence digraph for consistency with the companion missions; every statement here assumes it has no arcs.
• Implicit hypothesis. Theorems 1 and 5 assume every job fits alone ($r_{hj}\le s_h$), which the paper leaves unstated; without it no feasible schedule exists.
• The algorithm is a Lean definition following the page: the order is an argument (any nonincreasing order), "as early as possible" is the earliest start after $M_2$'s last job at which the resource constraint holds throughout, and the loop stops at the first move that does not strictly reduce $C_{\max}$.
• Optimality is always stated in full: feasibility plus a lower bound against every feasible schedule. No minimum is written as an infimum of a possibly empty set.
• Theorem 6 is stated with 0–1 slot assignments, the interpretation the paper gives to $x_{ijk}$; the page's constraint $\sum_{k=1}^{m}$ is read as $\sum_{k=1}^{n}$.
• Not formalized: the running times $O(ln^2+n^{5/2})$ (Theorem 1), $O(n\log n)$ (Theorem 5, including the phrase "This O(n log n) algorithm") and $O(n^3)$ (Theorem 6), which depend on a machine model the paper does not fix and, for Theorems 1 and 6, on cited matching and transportation algorithms.
• Ruled out: a formalization of the goal that proves optimality only against (a)–(c) schedules, against schedules with integer start times, or for one fixed tie-breaking order proves less than Theorem 5.

Contributions welcome: lemmas about step functions of resource usage on half-open intervals, a left-shifting lemma for unit-time schedules on two machines, and the exchange steps of Theorem 5 as separate lemmas.

Selected references

• J. Błażewicz, J.K. Lenstra, A.H.G. Rinnooy Kan, Scheduling subject to resource constraints: classification and complexity, Discrete Applied Mathematics 5 (1983) 11–24. https://doi.org/10.1016/0166-218X(83)90012-4
• M.R. Garey, D.S. Johnson, Complexity results for multiprocessor scheduling under resource constraints, SIAM Journal on Computing 4 (1975) 397–411. https://doi.org/10.1137/0204035
• R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics 5 (1979) 287–326. https://doi.org/10.1016/S0167-5060(08)70356-X
• S. Even, O. Kariv, An O(n^{2.5}) algorithm for maximum matching in general graphs, Proc. 16th IEEE FOCS (1975) 100–112. https://doi.org/10.1109/SFCS.1975.23

Motivation

Pontryagin's maximum principle gives necessary conditions for optimality in deterministic optimal control: along an optimal trajectory, the optimal control maximizes (or minimizes) a Hamiltonian built from an adjoint process. For a system driven by Brownian noise the analogous statement was open in full generality for two decades. The difficulty appears exactly when the diffusion coefficient depends on the control and the control domain is not convex, the situation of controlled volatility in finance, of controlled noise intensity in engineering, and of any problem whose admissible actions form a discrete or otherwise nonconvex set.

Shige Peng's 1990 paper (SIAM J. Control Optim. 28(4)) closed that case. It introduced the second-order adjoint process and a second-order variational inequality, and it is the starting point of the modern theory of stochastic Hamiltonian systems and of backward stochastic differential equations as a tool in control.

Timeline.

• 1972: Kushner obtains necessary conditions for diffusions whose diffusion coefficient does not depend on the control (SIAM J. Control 10).
• 1973–1978: Bismut introduces the adjoint equation as a linear backward stochastic differential equation and develops duality methods (SIAM Review 20).
• Early 1980s: Bensoussan and Haussmann prove maximum principles for convex control domains or control-independent diffusion, using the first-order adjoint equation only.
• 1990: Peng proves the general principle, with control-dependent diffusion and an arbitrary nonempty control domain (this mission). In the same year Pardoux and Peng prove existence and uniqueness for nonlinear backward SDEs (Systems Control Lett. 14).
• 1999: Yong and Zhou give a textbook account of the theory (Springer).

Setting

Let $(\Omega,\mathcal F,P)$ be a probability space carrying a standard $d$-dimensional Wiener process $B=(B^1,\dots,B^d)$, and let $\mathcal F^t=\sigma\{B(s);0\le s\le t\}$ be its natural filtration. Fix a horizon $T>0$, an initial state $x_0\in\mathbb R^n$ and a nonempty control domain $U\subseteq\mathbb R^k$. The data are

$$g:\mathbb R^n\times\mathbb R^k\to\mathbb R^n,\quad \sigma=(\sigma^1,\dots,\sigma^d),\ \sigma^j:\mathbb R^n\times\mathbb R^k\to\mathbb R^n,\quad l:\mathbb R^n\times\mathbb R^k\to\mathbb R,\quad h:\mathbb R^n\to\mathbb R .$$

An admissible control $v$ is a progressively measurable $U$-valued process with $\sup_{t\le T}E|v(t)|^m<\infty$ for every $m\ge1$. Its trajectory solves the state equation

$$dx(t)=g(x(t),v(t))\,dt+\sum_{j=1}^d\sigma^j(x(t),v(t))\,dB^j(t),\qquad x(0)=x_0,$$

and its cost is $J(v)=E\int_0^Tl(x(t),v(t))\,dt+E\,h(x(T))$. A pair $(y,u)$ is optimal when $J(u)\le J(v)$ for every admissible $v$.

Assumption (3): $g,\sigma,l,h$ are $C^2$ in $x$, jointly continuous in $(x,v)$ together with their first and second $x$-derivatives; $g_x,g_{xx},\sigma_x,\sigma_{xx},l_{xx},h_{xx}$ are bounded; and $g,\sigma,l_x,h_x$ grow at most like $C(1+|x|+|v|)$.

The Hamiltonian is $H(x,v,p,K)=l(x,v)+(p,g(x,v))+\sum_j(K_j,\sigma^j(x,v))$. The first-order adjoint process $(p,K)$ solves the backward equation

$$-dp=\Big[g_x^*p+\sum_j\sigma_x^{j*}K_j+l_x\Big]dt-\sum_jK_j\,dB^j,\qquad p(T)=h_x(y(T)),$$

and the second-order adjoint process $(P,Q)$, symmetric-matrix valued, solves

$$-dP=\Big[g_x^*P+Pg_x+\sum_j\sigma_x^{j*}P\sigma_x^j+\sum_j\sigma_x^{j*}Q_j+\sum_jQ_j\sigma_x^j+H_{xx}\Big]dt-\sum_jQ_j\,dB^j,\qquad P(T)=h_{xx}(y(T)),$$

with all coefficients evaluated along $(y(t),u(t))$ and both solutions adapted to $\mathcal F^t$.

Formalization targets

Goal: Theorem 3 (p. 975)

If $(y,u)$ is optimal, then adjoint processes $(p,K)$ and $(P,Q)$ exist in $L^2_{\mathcal F}$, solving the two equations above, such that for every $v\in U$, for almost every $\tau\in[0,T]$, almost surely,

$$H\big(y,v,p,K-P\sigma(y,u)\big)+\tfrac12\operatorname{tr}\big(\sigma\sigma^*(y,v)P\big)\ \ge\ H\big(y,u,p,K-P\sigma(y,u)\big)+\tfrac12\operatorname{tr}\big(\sigma\sigma^*(y,u)P\big),$$

all evaluated at time $\tau$.

Milestones

1. Lemma 1: the spike-perturbed state equals $y+y_1+y_2$ up to $o(\varepsilon^2)$ in mean square, where $y_1,y_2$ solve the first- and second-order variational equations (5), (6).
2. Lemma 2: the cost expansion (11) is $\ge o(\varepsilon)$ at an optimal control.
3. Eq. (13): existence and uniqueness of $(p,K)$ as a Riesz representer.
4. Eq. (14): the cost expansion in Hamiltonian form is $\ge o(\varepsilon)$.
5. Eq. (17): existence and uniqueness of $(P,Q)$ as a Riesz representer.
6. Eq. (18): the variational inequality for these representers.
7. Eq. (19): $(p,K)$ is the unique solution of the first-order adjoint equation.
8. Eq. (20): $(P,Q)$ solves the second-order adjoint equation.

Significance

The result. Theorem 3 is the necessary condition for optimal control of diffusions in its general form. When $\sigma$ does not depend on the control the trace terms cancel and it reduces to the classical first-order principle. When the control enters the diffusion, the first-order condition is false in general, and the second-order adjoint $P$ is the correction. The theorem underlies stochastic linear-quadratic theory, the verification of optimal portfolio and volatility-control policies, and the relation between the maximum principle and the Hamilton–Jacobi–Bellman equation.

Formalizing it. The theorem is classical and proved; none of it is machine-checked. Mathlib has real Brownian motion but no stochastic integral, no SDE and no backward SDE. This mission produces the first formal statements on the platform of a controlled SDE, of a backward SDE and of the maximum principle, together with a precise definition layer (Itô integral, Itô process, BSDE solution) that later missions can reuse, e.g. for Peng's endpoint-constrained principle (§6 of the paper) or for the existence theory of BSDEs. A formal proof would also pin down the approximation arguments the paper leaves to the reader.

Difficulty

The obvious route perturbs the optimal control convexly, $u+\varepsilon(v-u)$, and differentiates the cost. That needs $U$ convex. For nonconvex $U$ one uses a spike variation on a time interval of length $\varepsilon$. For deterministic systems the state then moves by $O(\varepsilon)$ and a first-order expansion suffices. With control-dependent diffusion the stochastic integral over the spike interval moves the state by order $\sqrt\varepsilon$ in $L^2$, so the first-order variational equation leaves an error of the same order as the effect being measured. Second-order terms in the state enter the cost at order $\varepsilon$, and they are quadratic, so they cannot be handled by a single linear adjoint. The second-order expansion, the matrix-valued adjoint that represents the quadratic term, and the identification of both adjoints with backward SDEs are where the work lies. On the formal side, none of the stochastic calculus exists in Mathlib: the Itô isometry, Itô's formula for matrix-valued processes, moment estimates for linear SDEs and the martingale representation behind the backward equations all have to be built.

Formalization scope

Conventions committed to in Lean:

• States, controls and noise are Fin n → ℝ, Fin k → ℝ, Fin d → ℝ with the sup norm; matrices are Matrix (Fin n) (Fin n) ℝ. Time is ℝ≥0, and time integrals are over [0, t] ⊂ ℝ.
• The Wiener process is $\mathbb R^d$-valued: the paper's "$R^n$-valued standard Wiener process" (p. 967) is a misprint, since $\sigma(x,v)\in\mathcal L(R^d,R^n)$. Coordinates are independent real Brownian motions (Mathlib's IsBrownianReal).
• The filtration is the natural filtration of $B$, not completed, as on p. 967.
• "Adapted", for processes integrated in $dt$, is read as progressively measurable.
• The Itô integral is a relation (an $L^2$ limit of elementary integrals, as in Ikeda–Watanabe), not an operator. SDE and BSDE solutions hold "for every $t$, almost surely", with $\sup_tE|x(t)|^2<\infty$ for forward solutions and $L^2_{\mathcal F}$ membership for backward ones.
• Optimality is among admissible controls of finite cost, and the optimal cost is finite; the paper never states finiteness, and a cost can be $+\infty$ under (3).
• Lemma 1 is stated with $o(\varepsilon^2)$ where the page prints "$\le C\varepsilon^2$" in (4). The proof (via (10)) establishes $o(\varepsilon^2)$, and Lemma 2 needs it. Lemma 1, (13), (17), (19) and (20) are stated for any admissible pair, since their proofs do not use optimality.
• "$\ge o(\varepsilon)$" means: some $r(\varepsilon)=o(\varepsilon)$ as $\varepsilon\to0^+$ bounds the left side from below for small $\varepsilon$. "$\forall v\in U$, a.e., a.s." quantifies $v$ first, then $\tau$, then $\omega$.
• $P$ and $Q_j$ are symmetric-valued ($R^{n,n}$ is the space of symmetric matrices, p. 973). Stochastic integrals against the matrix $\sigma$ or $Q$ are sums over the columns, $\sum_j(\cdot)_j\,dB^j$.

Trivializing formalizations ruled out. Without adaptedness the backward equations have pathwise solutions with $K=0$ and the goal would be free; every adjoint and every solution is required to be progressive for the natural filtration of $B$. The hypotheses are satisfiable: a sorry-free check shows that the zero problem has an optimal pair and that the Itô relation holds for the zero integrand.

Infrastructure needed, and reusable. The Itô integral and isometry, Itô's formula (vector and matrix forms), existence, uniqueness and moment estimates for linear SDEs with bounded coefficients, Riesz representation in $L^2_{\mathcal F}$, and existence and uniqueness for linear BSDEs (via martingale representation for the Brownian filtration). All of these are reusable well beyond this mission; contributions of any of them, as separate theorems, are welcome. Related platform definitions: the Ethier–Kurtz series (EthierKurtz_HasBrownianItoIntegral, EthierKurtz_SolvesBrownianSDE) formalizes an Itô integral by dyadic step approximation and uncontrolled SDEs over a completed filtration. The deterministic Pontryagin principle appears in Vector Space Methods XII and Dynamic Programming and Optimal Control III.

Selected references

• S. Peng, A General Stochastic Maximum Principle for Optimal Control Problems, SIAM J. Control Optim. 28(4), 966–979, 1990. https://doi.org/10.1137/0328054
• H. J. Kushner, Necessary Conditions for Continuous Parameter Stochastic Optimization Problems, SIAM J. Control 10(3), 550–565, 1972. https://doi.org/10.1137/0310041
• J.-M. Bismut, An Introductory Approach to Duality in Optimal Stochastic Control, SIAM Review 20(1), 62–78, 1978. https://doi.org/10.1137/1020004
• E. Pardoux and S. Peng, Adapted Solution of a Backward Stochastic Differential Equation, Systems & Control Letters 14(1), 55–61, 1990. https://doi.org/10.1016/0167-6911(90)90082-6
• J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, 1999. https://doi.org/10.1007/978-1-4612-1466-3

Motivation

Metrical task systems, introduced by Borodin, Linial and Saks (J. ACM 39(4), 1992), are a common abstraction of on-line problems in which a server occupies one of finitely many states, pays a cost for each task depending on its current state, and may pay a transition cost to change state first. Paging, list update and the $k$-server problem all fit into this framework. The paper's first main result is that every deterministic on-line algorithm on an $n$-state metrical task system has competitive ratio at least $2n-1$, and that $2n-1$ is attained. The lower bound comes from an adversary that always charges the state the algorithm currently occupies. That adversary needs to know the algorithm's state, which suggests that randomization can help.

Section 7 of the paper makes this precise for the simplest system, the uniform task system, in which all transitions cost $1$. There the randomized competitive ratio against an oblivious adversary is between $H(n)$ and $2H(n)$, where $H(n)=1+\tfrac12+\cdots+\tfrac1n$ is between $\ln n$ and $1+\ln n$. This was the first logarithmic bound for a task system.

Timeline:

• 1985: Sleator and Tarjan introduce competitive analysis for list update and paging (CACM 28(2)).
• 1987/1992: Borodin, Linial and Saks define metrical task systems, prove the deterministic ratio $2n-1$, and prove $H(n)\le\bar w\le 2H(n)$ for the uniform system (conference version STOC 1987; journal version cited above).
• 1991: Fiat, Karp, Luby, McGeoch, Sleator and Young prove the analogous $2H_k$ upper bound for randomized paging (J. Algorithms 12(4)).

Setting

A task system $(S,d)$ is a finite set $S$ of $n$ states with a transition-cost matrix $d$: $d(i,i)=0$, $d(i,j)>0$ for $i\ne j$, and $d(i,k)\le d(i,j)+d(j,k)$. In the uniform task system, $d(i,j)=1$ for all $i\neq j$. A task is a vector $T\in\mathbb R_{\ge0}^S$ of processing costs. Given an initial state $s_0$ and tasks $\mathbf T=T^1\cdots T^m$, a schedule is $\sigma:\{0,\dots,m\}\to S$ with $\sigma(0)=s_0$, of cost

$$c(\mathbf T;\sigma)=\sum_{i=1}^m d(\sigma(i-1),\sigma(i))+\sum_{i=1}^m T^i(\sigma(i)).$$

The off-line optimum $c_0(\mathbf T)$ is the least cost over all schedules.

A deterministic on-line algorithm chooses $\sigma(i)$ from $s_0$ and $T^1,\dots,T^i$. A randomized on-line algorithm $R$ chooses $\sigma(i)$ at random, with a distribution that depends on $s_0$, on $T^1,\dots,T^i$ and on the states $\sigma(0),\dots,\sigma(i-1)$ already visited. The task sequence is fixed before any random choice is made (an oblivious adversary). With $\mathrm{pr}(\sigma\mid\mathbf T)$ the probability that $R$ follows $\sigma$, the expected cost is $\bar c_R(\mathbf T)=\sum_\sigma c(\mathbf T;\sigma)\,\mathrm{pr}(\sigma\mid\mathbf T)$. For $w>0$, $R$ is expected $w$-competitive if there is a constant $K$ with

$$\bar c_R(\mathbf T)\le w\,c_0(\mathbf T)+K$$

for every finite task sequence and every initial state. The randomized competitive ratio $\bar w(S,d)$ is the infimum of all such $w$ over all $R$.

Formalization targets

Goal: Theorem 7.1

For the uniform task system on $n\ge1$ states,

$$H(n)\;\le\;\bar w(S,d)\;\le\;2H(n).$$

Milestones

1. Upper bound (p. 759). Some randomized on-line algorithm is expected $2H(n)$-competitive on the uniform task system.
2. Lemma 7.2 (p. 759). Let $D$ be a distribution on infinite task sequences over a finite task alphabet, with $E(c_0(\mathbf T^j))\to\infty$, and let $m_j=\inf_A E(c_A(\mathbf T^j))$ over deterministic on-line algorithms. Then every achievable $w$ satisfies

$$\limsup_{j\to\infty}\frac{m_j}{E(c_0(\mathbf T^j))}\le w .$$

3. $m_j\ge j/n$ (p. 760) when the tasks are independent uniformly random unit elementary tasks $U_s$ (cost $1$ in $s$, $0$ elsewhere).
4. Coupon collector (p. 760). For i.i.d. uniform states on $S$, the expected number of draws until every state has appeared is $nH(n)$.
5. Off-line cost (p. 760). Under the same distribution, $E(c_0(\mathbf T^j))\le j/(nH(n))+C$ for a constant $C$ independent of $j$.

Significance

The theorem shows that randomization reduces the competitive ratio of the uniform task system from $2n-1$ to $\Theta(\log n)$. That is an exponential improvement, and it identifies the adversary's knowledge of the algorithm's state as the source of the deterministic lower bound. Lemma 7.2 is a form of Yao's principle adapted to the additive-constant definition of competitiveness. It is the standard tool for randomized lower bounds in on-line computation, and the same argument shape reappears for paging and $k$-server lower bounds.

On the formalization side, the result is proved but, as far as is known, has not been machine-checked. A complete development yields a reusable model of randomized on-line algorithms with oblivious adversaries, a Yao-type lemma usable for other on-line problems, and a coupon-collector expectation in the product-measure setting. The upper half additionally needs the continuous-time reduction of the paper's Lemma 3.1 in randomized form, or a direct discrete-time algorithm.

Difficulty

For the upper bound, the natural algorithm is continuous-time. It proceeds in phases, and inside a phase it stays in a state until that state has accumulated cost $1$. A discrete task can saturate several states at once and straddle a phase boundary. So a discrete algorithm must either simulate the continuous one or be analyzed directly, and the expected transition count per phase must be controlled with the first phase starting in a deterministic state.

For the lower bound, the first obstacle is that the natural statement "$\bar w\ge\limsup m_j/E(c_0)$" silently assumes that a randomized algorithm's expected cost, averaged over random inputs, is at least that of the best deterministic algorithm. With the behavioural (kernel) definition used here, this requires converting a kernel into a mixture of deterministic algorithms, which is Kuhn's theorem on each finite horizon. The second obstacle is that the paper's claim $E(c_0(\mathbf T^j))\le j/(nH(n))+O(1)$ is supported only by the elementary renewal theorem, which gives a limit of ratios; the additive bound needs a sharper renewal estimate. Mathlib has no renewal theory. The hypothesis $E(c_0(\mathbf T^j))\to\infty$ of Lemma 7.2 must also be established for the uniform distribution; the paper does not prove it separately.

Formalization scope

• States form a finite nonempty type S, and $n$ = Fintype.card S; no $n\ge2$ assumption is made (at $n=1$ the goal reads $1\le\bar w\le2$, and $\bar w=1$). $H(n)$ is Mathlib's harmonic n, cast to $\mathbb R$. The uniform system has unit transition cost.
• Tasks are finite and nonnegative. The paper also allows $+\infty$ entries; these are excluded. Task sequences are Fin m → S → ℝ and schedules are Fin (m+1) → S with σ 0 = s₀. $c_0$ is a finite minimum.
• A randomized algorithm is a kernel S → List (S → ℝ) → List S → PMF S. This is the paper's scheduler–taskmaster description (p. 758), equivalent to a distribution over deterministic algorithms on every finite task sequence. $\mathrm{pr}(\sigma\mid\mathbf T)$ is the product of kernel probabilities, and $\bar c_R$ is a finite sum. That $\mathrm{pr}(\cdot\mid\mathbf T)$ sums to $1$ has been checked locally.
• $\bar w(S,d)$ is the real sInf of $\{w : \exists R,\ R \text{ expected } w\text{-competitive}\}$. On the empty set this would be $0$, so the upper bound is stated as the existence of an expected $2H(n)$-competitive algorithm, and Lemma 7.2 is stated for every achievable $w$. The goal's lower half forces the set to be nonempty. Statements of the form "$\bar w\le c$" alone are therefore not acceptable substitutes for milestones 1 and 2.
• Lemma 7.2 is restricted to task sequences over a finite alphabet, with the product $\sigma$-algebra and measurable singletons. This makes every $E(c_A(\mathbf T^j))$ a genuine integral for every deterministic $A$, and it covers the paper's application. The $\limsup$ of Lemma 7.2 is taken in EReal.
• The coupon-collector time takes values in $[0,\infty]$ and its expectation is a lower Lebesgue integral. Milestone 5 renders the paper's $O(1)$ as an explicit constant $C$ chosen before $j$.

Contributions welcome: proofs of any milestone; a discrete-time randomized phase algorithm; a general Kuhn-type conversion from kernels to mixtures of deterministic algorithms; renewal-theoretic lemmas.

Selected references

• A. Borodin, N. Linial, M. Saks, An Optimal On-Line Algorithm for Metrical Task System, J. ACM 39(4):745–763, 1992. https://doi.org/10.1145/146585.146588
• D. D. Sleator, R. E. Tarjan, Amortized Efficiency of List Update and Paging Rules, Commun. ACM 28(2):202–208, 1985. https://doi.org/10.1145/2786.2793
• A. Fiat, R. M. Karp, M. Luby, L. A. McGeoch, D. D. Sleator, N. E. Young, Competitive Paging Algorithms, J. Algorithms 12(4):685–699, 1991. https://doi.org/10.1016/0196-6774(91)90041-V
• A. C.-C. Yao, Probabilistic Computations: Toward a Unified Measure of Complexity, FOCS 1977, 222–227. https://doi.org/10.1109/SFCS.1977.24
• S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, 1970 (the elementary renewal theorem cited as [20] in the paper).

Motivation

Minimum mean square error (MMSE) estimation — predicting a signal $x$ from a noisy observation $y$ by minimizing expected squared prediction error — underlies linear systems theory, linear regression, Kalman filtering, and multiple-input multiple-output signal processing. Its classical solution assumes the joint distribution of $(x,y)$ is known exactly; in practice it is estimated from data, and the estimator inherits sampling error and model risk. Kuhn, Mohajerin Esfahani, Nguyen & Shafieezadeh-Abadeh's 2019 INFORMS TutORials chapter shows that hedging the MMSE objective against every distribution in a Wasserstein ball around the empirical distribution — an infinite-dimensional worst case over an intractable set of measures, a priori — collapses to a tractable, finite-dimensional convex semidefinite program (Theorem 25, p. 29), building on the Gelbrich-hull machinery of Section 2.3. This mission formalizes that reduction.

Setting

Fix $m_x, m_y \in \mathbb{N}$ and let $\xi = (x,y) \in \mathbb{R}^{m_x} \times \mathbb{R}^{m_y}$ be a random vector: $x$ the signal to be estimated, $y$ the observation. An estimator is a measurable function $\psi : \mathbb{R}^{m_y} \to \mathbb{R}^{m_x}$; write $\Psi$ for the family of all estimators. The distribution of $\xi$ is only known to lie in a type-2 Wasserstein ball $B_{\varepsilon,2}(\hat P_N)$ centered at an elliptical nominal distribution $\hat P_N = E_g(\hat\mu,\hat\Sigma)$ with nominal mean $\hat\mu \in \mathbb{R}^m$ ($m=m_x+m_y$), nominal covariance $\hat\Sigma \in S^m_+$, and density generator $g$. The distributionally robust MMSE estimation problem is

$$\inf_{\psi \in \Psi} \sup_{Q \in B_{\varepsilon,2}(\hat P_N)} E_Q\big[\|x-\psi(y)\|_2^2\big]. \tag{35}$$

Writing $\hat\Sigma = \begin{pmatrix}\hat\Sigma_{xx}&\hat\Sigma_{xy}\\\hat\Sigma_{yx}& \hat\Sigma_{yy}\end{pmatrix}$ blockwise, the nonlinear convex SDP

$$\max_S f(S) = \mathrm{Tr}[S_{xx} - S_{xy}S_{yy}^{-1}S_{yx}] \quad \text{s.t.} \quad S = \begin{pmatrix}S_{xx}&S_{xy}\\S_{yx}&S_{yy}\end{pmatrix} \succeq 0,\; S_{xx} \succeq 0,\; S_{yy} \succeq 0,\; \mathrm{Tr}[S+\hat\Sigma-2(\hat\Sigma^{1/2}S\hat\Sigma^{1/2})^{1/2}] \le \varepsilon^2,\; S \succeq \lambda_{\min}(\hat\Sigma) I \tag{36}$$

is the finite-dimensional relaxation the chapter builds toward.

Formalization targets

Goal (Theorem 25, distributionally robust MMSE estimator). If $\hat\Sigma \succ 0$, then the optimal value of problem (35) equals the optimal value of SDP (36). Moreover, if $S^\star$ is optimal in (36) with $S^\star_{yy}$ invertible, then the affine function

$$\psi^\star(y) = S^\star_{xy}(S^\star_{yy})^{-1}(y-\hat\mu_y) + \hat\mu_x$$

attains the outer infimum of (35) — it is a distributionally robust MMSE estimator, exhibited in closed form from an SDP optimizer.

Significance

Theorem 25 reduces an a priori infinite-dimensional, worst-case functional optimization problem (an infimum over all measurable estimators of a supremum over all distributions within a Wasserstein ball) to a finite convex program with one linear matrix inequality, one Loewner-order lower bound, and one trace/matrix-square-root constraint — solvable in polynomial time, with the optimal estimator recovered in closed form from the SDP's optimal block matrix. It shows that robustifying MMSE estimation against distributional ambiguity does not sacrifice tractability: the resulting estimator remains affine, the same functional form as the classical (non-robust) best linear unbiased estimator, only with its coefficients drawn from a regularized covariance estimate rather than the raw sample covariance. Formalizing it fixes, machine-checkably, the exact shape of that regularization — which SDP constraints are load-bearing (the Loewner lower bound in particular rules out a numerically unstable near-singular $S_{yy}$) and which conditions ($\hat\Sigma \succ 0$, $S^\star_{yy}$ invertible) the closed-form estimator formula actually needs.

Difficulty

The paper's own remark (p. 29) names the two nontrivial steps: first, "establishing a minimax theorem for (35) and exploiting the properties of elliptical distributions" to show the outer infimum is attained by an affine estimator — a priori (35) ranges over all measurable $\psi$, and there is no obvious reason the worst case forces linearity. Second, "combining this structural insight with Theorem 16" (the SDP-representability result for indefinite quadratic losses under an elliptical nominal distribution, itself a nontrivial closed-form reduction of an infinite-dimensional worst-case risk) to convert the now-restricted problem over affine estimators into the finite SDP (36). Neither step is a routine consequence of the ambiguity-set definitions alone; each requires structural facts about elliptical distributions and quadratic losses proved earlier in the chapter.

Formalization scope

The signal-observation space is EuclideanSpace ℝ (Fin mx ⊕ Fin my), with $x$ and $y$ recovered as the two summand projections; the block matrix $S$ is Matrix (Fin mx ⊕ Fin my) (Fin mx ⊕ Fin my) ℝ, and Matrix.toBlocks₁₁/toBlocks₁₂/toBlocks₂₁/toBlocks₂₂ give its four blocks. The constraint "$S_{xy}=S_{yx}^\top$" is not stated as a separate hypothesis: it follows automatically once $S$ is symmetric (implied by S.PosSemidef), so encoding the feasible set from a single symmetric S rather than four independently-quantified blocks makes it structurally impossible to drop — see pitfall 4 of BRIEF.md. The outer infimum of problem (35) ranges only over measurable $\psi$ (Measurable ψ on the binder), matching the paper's own definition of $\Psi$ as "the family of all possible measurable estimators" (p. 29) exactly. λ_min(Σ̂) is taken as a hypothesis parameter characterized by the two properties that make it the minimum ("≤ every eigenvalue of Σ̂, and attained by some eigenvalue"), rather than invoking a specific Mathlib min-eigenvalue API by name. S_{yy}⁻¹ uses the ordinary matrix inverse (junk zero matrix when singular), matching the paper's literal notation; S^\star_{yy} invertible is stated as an added hypothesis, not present verbatim on the page, because the paper leaves the formula's well-definedness implicit — disclosed per pitfall 5 rather than silently assumed away. The paper's own "which is always solvable" clause is not asserted: Theorem 25 states, as part of itself, that SDP (36) attains its maximum (an unconditional existence claim for an optimal $S^\star$); this formalization states only the conditional consequences of such an $S^\star$ existing, not that one does — proving or asserting solvability is out of this mission's scope, so the Lean statement is strictly weaker than Theorem 25's own conclusion on this point, disclosed rather than silently dropped. All risk-style suprema are EReal-valued and Integrable-guarded, matching the series' convention. No milestone theorem is included: the paper's own proof sketch derives (33)'s and by extension (36)'s SDP "via Theorem 16", but Theorem 16 (indefinite quadratic loss and $p=2$, eq. 23) was itself judged too heavy to state faithfully in 02-gelbrich's time budget and is not redefined here either — see STATUS.md. Theorem 24 (the Wasserstein shrinkage estimator, this chapter's originally recommended goal) is out of scope: its closed-form eigenvalue transformation (eq. 34a/34b) requires transcribing nested square roots from a rendered PDF page that this session's time budget did not allow verifying to the standard the brief demands (pitfall 1); the brief's own documented fallback to Theorem 25 was taken instead.

Selected references

• Kuhn, D., Mohajerin Esfahani, P., Nguyen, V. A., & Shafieezadeh-Abadeh, S. (2019). Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning. INFORMS TutORials in Operations Research. https://doi.org/10.1287/educ.2019.0198
• Nguyen, V. A., Shafieezadeh-Abadeh, S., Yue, M.-C., Kuhn, D., & Wiesemann, W. (2021). Optimistic distributionally robust optimization for nonparametric likelihood approximation. Advances in Neural Information Processing Systems, 32.
• Shafieezadeh-Abadeh, S., Nguyen, V. A., Kuhn, D., & Mohajerin Esfahani, P. (2018). Wasserstein distributionally robust Kalman filtering. Advances in Neural Information Processing Systems, 31.

Motivation

Distributionally robust optimization (DRO) hedges a decision against every distribution within some ambiguity set around an estimated (nominal) distribution, rather than trusting the estimate exactly. When the ambiguity set is a ball of radius $\varepsilon$ around the empirical distribution $\hat P_N$ in the type-$p$ Wasserstein metric, the resulting worst-case risk problem inherits attractive statistical guarantees (Mohajerin Esfahani & Kuhn 2018) but is, in general, an optimization problem over an infinite-dimensional space of measures. Kuhn, Mohajerin Esfahani, Nguyen & Shafieezadeh-Abadeh's 2019 INFORMS TutORials chapter surveys when this problem becomes computationally tractable. One route — the subject of this mission — discards everything about the nominal distribution except its mean vector and covariance matrix and replaces the Wasserstein ball with a set built only from these two moments, the Gelbrich hull. The construction is due to Gelbrich (1990), who first bounded the Wasserstein distance between two distributions using only their means and covariances.

Setting

Fix $\Xi \subseteq \mathbb{R}^m$, a nominal distribution $\hat P_N \in \mathcal{P}(\Xi)$, a radius $\varepsilon > 0$ and an exponent $p \ge 1$. The type-$p$ Wasserstein distance between two probability measures $Q, Q'$ on $\mathbb{R}^m$ is

$$W_p(Q,Q') = \Big(\inf_{\pi \in \Pi(Q,Q')} \int \|\xi-\xi'\|^p \, d\pi(\xi,\xi')\Big)^{1/p},$$

the infimum over couplings $\pi$ (probability measures on $\mathbb{R}^m \times \mathbb{R}^m$ with marginals $Q$ and $Q'$) of the $p$-th root of the expected $p$-th power of Euclidean distance. The Wasserstein ambiguity set is $B_{\varepsilon,p}(\hat P_N) = \{Q \in \mathcal{P}(\Xi) : W_p(Q,\hat P_N) \le \varepsilon\}$, and the worst-case risk of a loss function $\ell$ is $R_{\varepsilon,p}(\hat P_N,\ell) = \sup_{Q \in B_{\varepsilon,p}(\hat P_N)} E_Q[\ell(\xi)]$.

Suppose $\hat P_N$ has mean vector $\hat\mu$ and covariance matrix $\hat\Sigma \in S^m_+$ (the positive semidefinite $m\times m$ matrices). The mean-covariance uncertainty set is

$$U_\varepsilon(\hat\mu,\hat\Sigma) = \Big\{(\mu,\Sigma) \in \mathbb{R}^m \times S^m_+ : \|\hat\mu-\mu\|_2^2 + \mathrm{Tr}\big[\hat\Sigma+\Sigma-2(\hat\Sigma^{1/2}\Sigma\hat\Sigma^{1/2})^{1/2}\big] \le \varepsilon^2\Big\},$$

where $\Sigma^{1/2}$ is the positive-semidefinite square root. The Gelbrich hull is $G_\varepsilon(\hat\mu,\hat\Sigma) = \{Q \in \mathcal{P}(\Xi) : (E_Q[\xi],\mathrm{Cov}_Q[\xi]) \in U_\varepsilon(\hat\mu,\hat\Sigma)\}$: the distributions on $\Xi$ whose own mean and covariance lie in $U_\varepsilon(\hat\mu,\hat\Sigma)$. An elliptical distribution $E_g(\mu,\Sigma)$ has density $f(\xi) = C \cdot \det(\Sigma)^{-1} g\big((\xi-\mu)^\top\Sigma^{-1}(\xi-\mu)\big)$ for a density generator $g$ and normalizing constant $C$; two elliptical distributions "have the same density generator" when their $g$ coincide (e.g. both Gaussian, both Student-$t_\nu$ for the same $\nu$).

Formalization targets

Goal (Theorem 13, Gelbrich hull). For every $p \ge 2$,

$$B_{\varepsilon,p}(\hat P_N) \subseteq G_\varepsilon(\hat\mu,\hat\Sigma).$$

This is an outer approximation: every distribution within $\varepsilon$ of $\hat P_N$ in Wasserstein distance has a mean and covariance inside $U_\varepsilon(\hat\mu,\hat\Sigma)$, so optimizing over the Gelbrich hull instead of the Wasserstein ball can only enlarge the feasible set, never shrink it below the truth.

Supporting results. Theorem 4 (Gelbrich bound) gives the moment-only lower bound on $W_2$ that Theorem 13 is built from, with equality for elliptical distributions sharing a generator. Proposition 1 sharpens the goal's containment to an equality on the mean-covariance projection itself, under the same two conditions ($\Xi = \mathbb{R}^m$, $\hat P_N$ elliptical). Corollary 1 propagates the goal's set containment to the risk level: $R_{\varepsilon,p}(\hat P_N,\ell) \le R_\varepsilon(\hat\mu,\hat\Sigma,\ell)$ for every $\ell$, where $R_\varepsilon(\hat\mu,\hat\Sigma,\ell) = \sup_{Q \in G_\varepsilon(\hat\mu,\hat\Sigma)} E_Q[\ell(\xi)]$ is the Gelbrich risk.

Significance

Theorem 13 is the hinge between an intractable infinite-dimensional worst-case-risk problem and a tractable one: the paper goes on (Theorem 16, outside this mission's scope) to show that for quadratic loss functions and elliptical nominal distributions the Gelbrich risk itself equals the optimal value of a semidefinite program with two linear matrix inequality constraints — and that, under those same conditions, the Wasserstein worst-case risk, the Gelbrich risk and the SDP value all coincide. Corollary 1 is what makes the Gelbrich risk usable as a conservative surrogate even outside that special case: it upper-bounds the true worst-case risk for any loss function and any $p \ge 2$, at the cost of discarding all but first- and second-order information about the nominal distribution. Formalizing the goal and Corollary 1 gives the exact scope in which this moment-relaxation is licensed — the $p \ge 2$ restriction and the outer-approximation direction are both easy to get backwards, and this mission's Lean encoding fixes both irreversibly.

Difficulty

The obvious first argument is to prove containment pointwise: fix $Q \in B_{\varepsilon,p}(\hat P_N)$ and show its mean and covariance land in $U_\varepsilon(\hat\mu,\hat\Sigma)$. That reduces Theorem 13 to Proposition 1's containment half, which in turn reduces to the Gelbrich bound (Theorem 4) applied to the pair $(Q,\hat P_N)$ — the inequality direction of Theorem 4 suffices for containment; only the sharper equality direction (needed for Proposition 1's own equality clause) requires the elliptical hypothesis. The non-obvious step is Theorem 4 itself: bounding $W_2(Q,Q')$ below by a closed-form expression in the two distributions' first two moments only, for arbitrary $Q,Q'$ with those moments, requires an argument that survives every coupling $\pi$ — the paper's proof goes through a lower bound on the coupling's cross-covariance term via the eigenvalues of $\Sigma^{1/2}\Sigma'\Sigma^{1/2}$, not a direct manipulation of $W_2$'s definition.

Formalization scope

$\mathbb{R}^m$ is EuclideanSpace ℝ (Fin m); a "distribution" is a MeasureTheory.Measure on it constrained by Q Set.univ = 1 (probability) and Q Ξᶜ = 0 (support in Ξ). The Wasserstein distance is ENNReal-valued (Definition 1's infimum over couplings, matching 01-duality's convention); the worst-case and Gelbrich risks are EReal-valued suprema restricted to loss functions integrable under the candidate distribution, avoiding Mathlib's junk value for a non-integrable Bochner integral. $\Sigma^{1/2}$ is the positive-semidefinite matrix square root, picked by choice from its defining existential and applied in this mission only to matrices hypothesized (or, per Section 2.3's standing assumption, given) positive semidefinite. Elliptical distributions (IsElliptical) are represented by the paper's own density formula (a measure equal to volume.withDensity of C·det(Σ)⁻¹·g((ξ-μ)ᵀΣ⁻¹(ξ-μ)) for some C>0), together with the mean/covariance facts every theorem in this chunk reads off directly; an earlier draft kept only the latter, under which "same density generator" held vacuously for any moment-matched pair — corrected after moderation flagged it (see STATUS.md). Because 01-duality (Wasserstein distance, ambiguity set, worst-case risk) is not yet a published mission, this chunk redefines those objects locally in its own namespace rather than importing an unpublished draft, per the series' definition-reuse policy; a future upload can retire the duplication once 01-duality is live. A formalization that dropped Theorem 4's "same density generator" condition from its equality clause, or that stated the goal's containment for all $p\ge 1$ rather than $p \ge 2$, would be trivializing or simply false — both are explicit hypotheses in the Lean statements. Matrix.PosSemidef and its Loewner order carry the $S^m_+$ constraints; no elliptical-distribution or Gelbrich-hull infrastructure exists elsewhere on the platform, so this mission's definitions are original contributions reusable by any later extension (Theorem 16/17, Lemma 1/2's SDP representations) of this series.

Selected references

• Kuhn, D., Mohajerin Esfahani, P., Nguyen, V. A., & Shafieezadeh-Abadeh, S. (2019). Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning. INFORMS TutORials in Operations Research. https://doi.org/10.1287/educ.2019.0198
• Gelbrich, M. (1990). On a formula for the L2 Wasserstein metric between measures on Euclidean and Hilbert spaces. Mathematische Nachrichten, 147(1), 185–203.
• Mohajerin Esfahani, P., & Kuhn, D. (2018). Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Mathematical Programming, 171(1), 115–166.

Motivation

Most stochastic programs cannot be solved exactly: the expectation defining the objective is an integral over a continuous or high-dimensional random parameter, and evaluating it exactly is as hard as the optimization itself. The standard remedy is Monte Carlo: draw a sample of size ν from the random parameter, replace the true expectation by the sample average, and solve the resulting finite-dimensional "sample average approximation" (SAA) instead. This only helps if the SAA's optimal value and optimal solution actually converge to the true problem's as ν → ∞, and if that convergence is fast enough to be useful with a sample size one can actually draw and solve. Birge & Louveaux's Chapter 9, §9.5, states the two central asymptotic results that justify this approach for a general (not necessarily linear, not necessarily two-stage) stochastic program: a central limit theorem describing the SAA optimal value's fluctuations around the truth (Theorem 6, after Shapiro [1991]), and an exponential-rate large-deviation bound on how quickly both the SAA value and the SAA solution concentrate near their true counterparts as the sample grows (Theorem 7, after Dai, Chen & Birge [2000]). This mission formalizes both statements.

Setting

Fix a feasible set X ⊆ ℝⁿ of first-stage decisions and an outcome space Ξ carrying a σ-algebra. The book considers the general stochastic program

z* = inf_{x ∈ X} ∫_Ξ g(x,ξ) P(dξ),                                                        (5.1)

with g : ℝⁿ × Ξ → ℝ an abstract integrand — no longer specialized to the two-stage recourse cost Q(x,ξ) of Chapters 3–7, matching the book's own level of generality at this point in §9.5 — and ξ a random element of (Ξ, 𝓑, P). Given an i.i.d. sample ξ₁, ξ₂, … from P, the sample average approximation of size ν is

zν = inf_{x ∈ X} (1/ν) Σᵢ₌₁^ν g(x,ξᵢ) .                                                    (5.2)

Both z* and zν are attained (an optimal solution x* of (5.1); a random optimal solution xν(ω) of (5.2) at each sample outcome ω). The two chapter results describe, in different regimes, how (zν, xν) relates to (z*, x*) as ν → ∞.

Formalized in this mission: X sits in EuclideanSpace ℝ (Fin n); the sample is a sequence ξ : ℕ → Ω → Ξ on an ambient probability space (Ω, P), independent and identically distributed (Mathlib's iIndepFun/IdentDistrib); z*, x*, zν, xν are given as hypotheses that pin them down as the optimal value and an optimal point of (5.1)/(5.2) (a lower bound over X plus attainment at the named point), rather than computed via sInf/sSup of an image set — Real's extended-real infimum returns the junk value 0 on an unbounded-below or empty set, which would silently misstate the theorems if X, g are only assumed as loosely as the book states them.

Formalization targets

Goal — Chapter 9, Theorem 7 (p. 412)

∀ ε>0, ∃ α>0, ∃ β>0,
  (∀ ν>0, P[|zν − z*| ≥ ε] ≤ α·e^{−βν})
  ∧ (x* the unique optimal solution of (5.1) → ∀ ν≥1, P[‖xν − x*‖ ≥ ε] ≤ α·e^{−βν})

under the moment hypothesis: there exist a>0, θ₀>0, η : Ξ → ℝ with |g(x,ξ)| ≤ a·η(ξ) for all x ∈ X, and E[e^{θ·η(ξ)}] < ∞ for every θ ∈ [0,θ₀]. This is the mission's goal because it is a clean, self-contained existential-constants statement — no algorithm to define, unlike most of the chapter's other convergence results — and because, like Chunk 03's Theorem 6(a), the book's own proof cites an external paper (Dai, Chen & Birge [2000], Theorems 3.1–3.2) and gives no in-text derivation: the statement itself, not a derivation from a preceding numbered result of this book, is the mission's content.

Milestone — Chapter 9, Theorem 6 (p. 411)

X compact, g(x,·) measurable ∀x∈X, g Lipschitz in x with an L²(μ) envelope a,
x0 the unique minimizer of x ↦ E g(x) over X
  ⟹ √ν·[zν − E g(x0)] converges in distribution to N(0, Var g(x0))

Included as a milestone (not used in Theorem 7's proof, which the book does not give — see above) because it is the chapter's other general SAA convergence result, standing on the same setup (5.1)–(5.2), and because Mathlib's MeasureTheory.Function.ConvergenceInDistribution (the TendstoInDistribution predicate) plus Probability.Distributions.Gaussian.Real (gaussianReal) and Mathlib's own i.i.d. central limit theorem (ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum_sub) supply exactly the vocabulary needed to state — not prove — a faithful weak-convergence-to-Gaussian conclusion. Chunk 09's BRIEF.md flagged this as a milestone to attempt "only if your workspace has enough of a weak-convergence/CLT toolkit in Mathlib to state it faithfully"; the toolkit is present (verified directly, not assumed from substrate.md, which predates this rev's addition of ConvergenceInDistribution.lean), so it is included.

Significance

Every practical Monte Carlo solution method for stochastic programming — every discretization, every scenario-reduction heuristic, every "solve on a sample and hope" approach used throughout the rest of the book and the wider literature — rests on exactly these two results: that the SAA converges at all (Theorem 6's CLT gives the asymptotic distribution of the error) and that it converges fast enough to bound the error at a finite, computable sample size (Theorem 7's exponential rate). Formalizing them gives Prove2Me a first foothold in convergence-rate theory for stochastic optimization under sampling, a genre distinct from the concentration-of-measure results already reachable via Mathlib's sub-Gaussian machinery (Probability/Moments/SubGaussian.lean): sub-Gaussian concentration bounds a fixed-size sample's deviation from its own mean, not the rate-in-ν convergence of a nested sequence of optimization problems' values and solutions to a limiting problem's — the object Theorem 7 is actually about.

Difficulty

Theorem 6 needs a functional/uniform argument over the whole feasible set X (not the plain i.i.d. CLT at the single point x0) to control the interaction between sampling noise and the optimization over x; the book states it without proof, citing Shapiro [1991]. Theorem 7's constants α, β are produced by a large-deviation argument specific to the exponential-moment condition, again cited rather than derived in the book. Both are left as sorry; the value of this mission is the faithful statement, matching the difficulty pattern already established for Chunk 03's Theorem 6(a) (a result the book itself only cites).

Formalization scope

• Existential constants left abstract, never sharpened or weakened. Theorem 7's α, β are ∃-bound exactly as the book leaves them (trap 8 of reference/FAITHFULNESS_TRAPS.md: the existentials sit outside every quantifier they must be uniform over — in particular outside the ∀ ν). No closed form for α, β in terms of a, θ0, ε is invented.
• The book's own typo is corrected, and the correction is flagged. The printed (5.8) reads P[E[zν − z*)] ≥ ε] ≤ αe^{−βν} — an unmatched parenthesis and a stray E[·] around a quantity that is already deterministic. milestones.yaml/MODERATION_NOTES.md quote the typo verbatim; the Lean and natural_language_statement use the unambiguous P[|zν − z*| ≥ ε] the surrounding prose (and every other occurrence of this quantity in the section) plainly intends.
• g is left fully abstract, not specialized to the two-stage recourse cost Q(x,ξ) of Chunks 03–07, matching §9.5's own generality and keeping this mission independent of every other chunk's namespace (no cross-chunk import, per missions/README.md's "Prior art" column for this chunk: "none expected").
• z*, x*, zν, xν are hypothesis-characterized, not sInf/sSup-defined, to avoid the real extended-value junk-value trap (trap 5) discussed under Setting above.
• Measurability of zν, xν is an added hypothesis (hzSAA_meas/hxSAA_meas/hzSAA_meas in Theorem 6), not derivable from the other hypotheses since g is abstract; the book is silent on this technical point, standard for an applied convergence theorem, but Lean's P {ω | …} needs it for the displayed probability to be the actual measure of the event rather than an outer-measure value on a possibly non-measurable set.
• Convergence in distribution (Theorem 6) is formalized via Mathlib's TendstoInDistribution, with the limiting Gaussian supplied as an explicit random variable Y on a separate probability space with HasLaw Y (gaussianReal 0 σ²) P' — the same pattern Mathlib's own CLT (tendstoInDistribution_inv_sqrt_mul_sum_sub) uses for its own conclusion.
• Trivialization risk (this chapter's own, beyond paper.md's book-wide list item 5). A formalization that quantifies α, β universally, or with an invented closed form, would assert something the book's proof (cited, not given) does not establish; a formalization of Theorem 7 that used a computable sInf-defined zν on a set that is not shown bounded below would let the conclusion hold vacuously via the junk value 0, independent of the genuine large-deviation content — both are excluded by the choices above.

Selected references

• Birge, J.R., Louveaux, F. Introduction to Stochastic Programming, 2nd ed., Springer 2011, Chapter 9, §9.5 (pp. 409–412).
• Shapiro, A. "Asymptotic properties of statistical estimators in stochastic programming." Annals of Statistics 19 (1991), 1463–1466 — proof of Theorem 6 (their Theorem 3.3).
• Dai, L., Chen, C.-H., Birge, J.R. "Convergence properties of two-stage stochastic programming." Journal of Optimization Theory and Applications 106 (2000), 489–509 — proof of Theorem 7 (their Theorems 3.1–3.2).
• King, A.J., Rockafellar, R.T. "Asymptotic theory for solutions in statistical estimation and stochastic programming." Mathematics of Operations Research 18 (1993), 148–162 — the general theory of §9.5's opening (Theorem 5), the chapter's third general result, not formalized here (see STATUS.md for why it is out of scope).