Optimization

Why airlines protect seats

An airline sells the seats of one flight in several fare classes at different prices, all drawn from one shared cabin. Discount fares are bought early, under advance-purchase restrictions, while most high-fare requests arrive close to departure. Accepting every early low-fare request fills the aircraft with cheap passengers and turns away late high-fare passengers; refusing too many leaves seats empty. Seat inventory control decides how many seats to keep away from the low fare. Peter Belobaba's 1987 MIT thesis (Flight Transportation Laboratory Report R87-7) introduced the expected marginal seat revenue (EMSR) model for this decision, and EMSR-type rules remain the basis of the booking-limit logic in airline revenue management systems.

Timeline. Littlewood (1972, AGIFORS Symposium Proceedings; reprinted 2005) proposed accepting a low-fare request as long as its fare is at least the high fare times the probability of selling all remaining seats to high-fare passengers. Analysts at Trans World Airlines (1973) and Richter at Lufthansa (1982) gave equivalent formulations for the dynamic case. Belobaba (1987, Ch. 5) restated the two-class rule as a static protection level for nested inventories and extended it heuristically to many classes. Brumelle and McGill (Operations Research 41, 1993) and Curry (Transportation Science 24, 1990) later proved optimality of nested protection levels for any number of classes under low-to-high arrivals, and showed that Belobaba's multi-class EMSR levels are not optimal for three or more classes. This mission concerns only the two-class result, which is correct.

Setting

A single flight leg has capacity $C \in \mathbb N$. Class 1 has fare $f_1$, class 2 has fare $f_2$, with $0 \le f_2 \le f_1$. The numbers of requests for the two classes are random variables $r_1, r_2$ with values in $\mathbb N$, defined on a probability space $(\Omega, \mu)$ and independent. There are no cancellations, no no-shows, and a refused request is lost.

The inventory is nested: a class-1 request is accepted as long as any seat is unsold. A protection level $S \in \{0, \dots, C\}$ is the number of seats reserved for class 1; it sets the class-2 booking limit $BL_2 = C - S$. All class-2 requests arrive before any class-1 request. Class 2 therefore books $\min(r_2, C - S)$ seats and class 1 books $\min(r_1, C - \min(r_2, C-S))$, and the realised revenue is

The expected revenue is $\bar R(S) = \mathbb E[R_S]$.

The tail probability of class 1 is $\bar P_1(S) = P[r_1 \ge S]$, the probability of receiving $S$ or more class-1 requests, and the expected marginal seat revenue of the $S$-th class-1 seat is

For a single class with $S$ seats the expected revenue is $f_1\,\mathbb E[\min(r_1, S)]$, and $\mathrm{EMSR}_1(S)$ is its increment from $S-1$ to $S$ seats. The EMSR protection level $S_2^1$ is the largest integer $S \in \{0, \dots, C\}$ with

In Lean these objects are nestedRevenue, expectedNestedRevenue, tailProb, classRevenue, emsr and emsrProtectionLevel in SeatInventory.Nested.

Formalization targets

Goal: Eqs. (5.15)–(5.16), optimality of the EMSR protection level

The goal fixes no distribution: it holds for every pair of independent $\mathbb N$-valued demands, and $S_2^1$ depends only on $f_2/f_1$ and the law of $r_1$.

Milestones

1. Eq. (5.11). $f_1\mathbb E[\min(r_1,S)] - f_1\mathbb E[\min(r_1,S-1)] = f_1 P[r_1 \ge S]$ for $S \ge 1$.
2. Eqs. (6.1)–(6.2). $\bar P_1$ and, for $f_1 \ge 0$, $\mathrm{EMSR}_1$ are non-increasing in $S$.
3. Eq. (4.8), Littlewood's rule, already on the platform as RevenueManagement.littlewood_marginal_value (Talluri and van Ryzin's Eq. (2.1), proved).
4. Sect. 5.2, p. 112. $\bar R(S) \le \bar R(S_2^1)$ for $S_2^1 \le S \le C$: a smaller booking limit for class 2 cannot raise expected revenue.
5. Sect. 5.2, p. 114. With the same class-2 limit $C - S$, the expected nested revenue is at least the expected revenue of two distinct inventories with $S$ and $C - S$ seats, strictly if $f_1 > 0$ and $P[r_2 < C - S,\ r_1 > S] > 0$.

Significance

The two-class result says that, for a static booking limit set once before sales open and low-fare demand arriving first, the airline needs only the high-fare demand distribution and the fare ratio to set the optimal limit; the low-fare forecast is irrelevant. This is the rule that the thesis then applies class by class in multi-class nested systems, and it is the base case against which the later exact multi-class theory (Brumelle–McGill, Curry) is checked. Milestone 5 makes precise why nested inventories dominate the distinct-inventory allocation of the thesis's Sect. 5.1 with the same class-2 limit.

The result is classical and proved, in the sense that the optimality of a two-class threshold policy follows from Littlewood's argument and from the dynamic-programming treatment in Talluri and van Ryzin's The Theory and Practice of Revenue Management (2004, Ch. 2). On Prove2Me, Littlewood's marginal rule and the dynamic-programming optimality of nested protection levels (RevenueManagement.static_optimal_controls) are formalized, but in Bellman form: there the protection level is defined through the value function of a dynamic program. What is not formalized is the statement in Belobaba's form, where the protection level is the explicit threshold of $f_1 P[r_1 \ge S]$ against $f_2$ and the objective is the explicit expected revenue of a booking limit. Connecting the two forms, and the comparison with distinct inventories, is the work of this mission.

Difficulty

The expected revenue couples the two demands through the capacity left by class 2, so $\bar R$ is not a sum of single-class revenues and is not separately concave in an obvious way. The step that requires care is the increment $\bar R(S) - \bar R(S-1)$: it is not $\mathrm{EMSR}_1(S) - f_2$, as the thesis's sentence after the milestone on p. 112 suggests, because the extra protected seat matters only on the event that class 2 would have reached its limit. Independence of $r_1$ and $r_2$ is what makes that event's probability factor out; without independence the threshold rule is not optimal. The discrete reading matters too: with $P[r_1 > S]$ in place of $P[r_1 \ge S]$ the rule is off by one seat and the claim fails.

Formalization scope

Conventions the Lean statements commit to:

• Demands are $\mathbb N$-valued measurable random variables r₁ r₂ : Ω → ℕ on a probability space μ; the goal and milestone 4 assume IndepFun r₁ r₂ μ. The thesis writes continuous densities (Eqs. (5.1)–(5.5)) but requires integer seat counts; the discrete model is used throughout.
• $\bar P_1(S) = P[r_1 \ge S]$, as in Eq. (6.2) and the prose of Eq. (5.11), not $P[r_1 > S]$ as in Eq. (5.2).
• The EMSR protection level is the largest $S \in \{0,\dots,C\}$ with $f_1 P[r_1 \ge S] \ge f_2$ (Eq. (5.15)); Eq. (5.16)'s equality is the continuous idealisation and is not stated.
• Booking order: all class-2 requests precede all class-1 requests (pp. 108, 112). This order is built into the revenue formula, not assumed separately.
• Fares satisfy $0 \le f_2 \le f_1$; the thesis has $f_1 > f_2$, and the statements also cover equality.
• Expectations are Bochner integrals of bounded revenues, probabilities are μ.real; seat counts use truncated subtraction only where $S \le C$.

A trivializing formalization is ruled out: $S_2^1$ is defined by the threshold of (5.15), never as an argmax of expected revenue, and the expected revenue is computed from the realised revenue of the booking process, not postulated as a sum of marginal terms.

The multi-class EMSR levels of Eqs. (5.19)–(5.29) and the dynamic revision of Eqs. (5.31)–(5.32) are out of scope. Proofs need the discrete expectation identity $\mathbb E[\min(r,S)] - \mathbb E[\min(r,S-1)] = P[r \ge S]$ and expectation of products of independent bounded functions, both in Mathlib's reach and reusable for other single-leg revenue models. Proofs of any milestone, and a proof of the goal from milestones 1, 2 and 4 plus the matching lower-half argument, are welcome.

Selected references

• P. P. Belobaba, Air Travel Demand and Airline Seat Inventory Management, PhD thesis, MIT, Flight Transportation Laboratory Report R87-7, 1987 (no DOI).
• K. Littlewood, Forecasting and control of passenger bookings, AGIFORS Symposium Proceedings 12, 1972; reprinted in Journal of Revenue and Pricing Management 4, 2005. https://doi.org/10.1057/palgrave.rpm.5170134
• S. L. Brumelle and J. I. McGill, Airline seat allocation with multiple nested fare classes, Operations Research 41, 1993. https://doi.org/10.1287/opre.41.1.127
• R. E. Curry, Optimal airline seat allocation with fare classes nested by origins and destinations, Transportation Science 24, 1990. https://doi.org/10.1287/trsc.24.3.193
• K. T. Talluri and G. J. van Ryzin, The Theory and Practice of Revenue Management, Springer, 2004. https://doi.org/10.1007/b139000

Motivation

The simplex method is the standard algorithm for linear programming, and its correctness rests on one question: does it stop? Each pivot of the method moves from one dictionary to another without decreasing the objective value, but a pivot can leave the objective unchanged. When that happens repeatedly, the method can return to a dictionary it has already visited and loop forever. This behaviour, cycling, is not hypothetical: Vanderbei's Chapter 3 exhibits a problem with four decision variables and three constraints on which the "largest coefficient" entering rule with a natural tie-breaking rule cycles through six dictionaries (Vanderbei 2014, pp. 26–27).

The chapter answers the question with two pivoting rules under which the simplex method provably terminates, and then draws the consequence that makes linear programming a finite theory: the fundamental theorem of linear programming. This mission formalizes the chapter's four numbered theorems in Vanderbei's own setting of standard-form problems with slack variables.

Timeline. Hoffman (1953) and Beale (1955) gave the first examples of cycling. The perturbation and lexicographic methods go back to Charnes (1952) and to Dantzig, Orden and Wolfe (1955). Bland (1977) introduced the smallest-index rule and proved that the simplex method terminates under it (Bland 1977).

Setting

A linear program in standard form has $m$ constraints and $n$ decision variables:

A solution $x$ is feasible if it satisfies every constraint, and optimal if in addition it maximizes the objective among feasible solutions. The problem is infeasible if no feasible solution exists, and unbounded if it has feasible solutions with arbitrarily large objective values.

The slack variables $w_i = b_i - \sum_j a_{ij}x_j$ are appended to the list of variables as $x_{n+i} = w_i$, so that the constraints become the linear system $[A\ I]\,x = b$ with $x \ge 0$ in $\mathbb{R}^{n+m}$. A dictionary is given by a set $\mathcal B$ of $m$ basic indices whose columns of $[A\ I]$ are linearly independent; the remaining indices $\mathcal N$ are nonbasic. Solving for the basic variables gives

The basic solution of the dictionary sets the nonbasic variables to zero. The dictionary is feasible if $\bar b_i \ge 0$ for every $i\in\mathcal B$, and degenerate if $\bar b_i = 0$ for some $i\in\mathcal B$.

The simplex method (Phase II) starts at a feasible dictionary and repeats a pivot: an entering variable $x_k$ is chosen among the nonbasic variables with $\bar c_k > 0$, and a leaving variable $x_l$ among the basic variables with $\bar a_{lk} > 0$ that minimize the ratio $\bar b_l/\bar a_{lk}$; then $x_k$ becomes basic and $x_l$ nonbasic. The method stops when no $\bar c_j$ is positive (the dictionary is optimal) or when the entering column has no positive $\bar a_{ik}$ (the problem is unbounded). A pivoting rule resolves the remaining choices. Bland's rule chooses both the entering and the leaving variable as the candidate with the smallest index. The lexicographic rule perturbs the right-hand sides by symbols $0<\epsilon_m\ll\dots\ll\epsilon_1\ll$ all data and chooses the leaving variable by the perturbed ratio test.

Formalization targets

Goal: Theorem 3.3 (termination under Bland's rule, p. 31)

From every feasible dictionary $D_0$, there is no infinite sequence of pivots

in which both the entering and the leaving variable follow Bland's rule; and a finite sequence of such pivots reaches a dictionary $D_T$ at which the method stops, optimal or exhibiting unboundedness.

Milestones

• Theorem 3.1 (p. 27): if the simplex method fails to terminate, it must cycle, i.e. an infinite run visits some dictionary twice.
• Theorem 3.2 (p. 30): the simplex method always terminates when the leaving variable is selected by the lexicographic rule.
• Theorem 3.4 (p. 33), the fundamental theorem: (1) with no optimal solution the problem is infeasible or unbounded; (2) if a feasible solution exists, a basic feasible solution exists; (3) if an optimal solution exists, a basic optimal solution exists.

Significance

The result. Termination under Bland's rule is what turns the simplex method from a heuristic into an algorithm. Combined with Phase I, it yields the fundamental theorem of linear programming, which reduces the search for an optimum to finitely many basic solutions and underlies the duality theory of the following chapters. Bland's rule also needs no perturbation or extra bookkeeping, and it is the anticycling rule used in many correctness proofs of simplex-type and combinatorial pivoting algorithms, including oriented-matroid programming.

Formalizing it. All four theorems are classical and proved. The Prove2Me library has the lexicographic rule and a nondegenerate termination theorem in the tableau setting of Bertsimas and Tsitsiklis (equality form $Ax=b$, $x\ge 0$, full row rank), and the existence of basic feasible and optimal solutions in that form. It has no statement of Bland's theorem, and none of Vanderbei's dictionary formulation over $[A\ I]$. This mission produces a machine-checkable model of dictionaries and pivoting rules in that formulation, and targets Bland's theorem, whose proof is a genuine combinatorial argument rather than a monotonicity argument.

Difficulty

The natural argument for termination is monotonicity: each pivot increases the objective, so no dictionary repeats. It fails exactly at degenerate pivots, where the step length $\bar b_l/\bar a_{lk}$ is zero and the objective and the basic solution do not change. Bland's rule gives no potential function that strictly increases along degenerate pivots, so the proof has to reason about a hypothetical cycle as a whole: which variables enter and leave the basis within it, and how two dictionaries of the cycle, in which the same variable leaves and later enters, constrain each other's coefficients. Relating the coefficients of two different dictionaries of the same problem is the step that has no counterpart in the model's definitions and has to be developed.

For Theorem 3.2, the symbols $\epsilon_i$ cannot be replaced by a fixed small real number: the method treats them as formal quantities on separate scales, and the statement is about that symbolic rule.

Formalization scope

Vectors are Fin n → ℝ, Fin m → ℝ, and the constraint matrix is Matrix (Fin m) (Fin n) ℝ. The $n+m$ variables are indexed by Fin (n + m) with the decision variables first and the slacks after them, which is the order $x_1,\dots,x_n,x_{n+1}=w_1,\dots,x_{n+m}=w_m$ that Bland's rule compares. A dictionary is a structure holding its basic set, a proof that it has $m$ elements and a proof that its columns of $[A\ I]$ are linearly independent; the coefficients $\bar b,\bar a,\bar c,\bar\zeta$ are computed as coordinates in the basis of basic columns. A dictionary is therefore determined by its basic set, as the proof of Theorem 3.1 uses. "Basic solution" is defined through such a dictionary, not as a support condition.

Termination is stated as the nonexistence of an infinite run from a feasible dictionary, for Theorems 3.2 and 3.3. The goal adds that a finite Bland run reaches a stopping dictionary, so that it cannot hold because pivots fail to exist. The lexicographic rule is encoded by lexicographic comparison of the coefficient vectors $(\bar b_i, r_{i1},\dots,r_{im})/\bar a_{ik}$ of the perturbed ratios; the symbol $\epsilon_p$ is attached in the starting dictionary to its $p$-th basic variable in increasing index order, which for the initial dictionary is the $p$-th constraint. Unboundedness in Theorem 3.4 is "for every $M$ a feasible solution with objective $>M$", as defined on p. 7. The statements carry no explicit constants.

A formalization that stated Theorem 3.3 for arbitrary pivot sequences with pairwise distinct bases would be Theorem 3.1's counting argument, not Bland's theorem; the goal is stated for pivots that follow Bland's rule and only those.

A complete development needs: the pivot update of a dictionary and the invariance of the solution set under it, feasibility preservation by the ratio test, the relation between the objective rows of two dictionaries, and finiteness of the set of bases. These are reusable for any later formalization of simplex-type algorithms. Proofs of the milestones, alternative proofs of Theorem 3.3, and Phase I (to connect Theorem 3.4 with the algorithm) are welcome.

Selected references

• R. J. Vanderbei, Linear Programming: Foundations and Extensions, 4th ed., International Series in Operations Research & Management Science 196, Springer, 2014, Chapter 3. https://doi.org/10.1007/978-1-4614-7630-6
• R. G. Bland, New finite pivoting rules for the simplex method, Mathematics of Operations Research 2(2):103–107, 1977. https://doi.org/10.1287/moor.2.2.103
• G. B. Dantzig, A. Orden, P. Wolfe, The generalized simplex method for minimizing a linear form under linear inequality restraints, Pacific Journal of Mathematics 5(2):183–195, 1955. https://doi.org/10.2140/pjm.1955.5.183
• E. M. L. Beale, Cycling in the dual simplex algorithm, Naval Research Logistics Quarterly 2(4):269–275, 1955. https://doi.org/10.1002/nav.3800020406
• D. Bertsimas, J. N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, 1997, §3.4 (lexicographic rule and Bland's rule in tableau form).

Motivation

Many optimization problems in operations research have objectives that are convex but not differentiable: the maximum of finitely many linear or smooth functions, the value function of a Lagrangian dual, the cost of a two-stage linear program as a function of the first-stage decision. Gradient methods do not apply to these directly. N. Z. Shor's Minimization Methods for Non-Differentiable Functions (Springer Series in Computational Mathematics 3, 1985; translated by K. C. Kiwiel and A. Ruszczyński from the 1979 Russian edition) develops the algorithms that replace the gradient by a subgradient, and its first chapter sets up the calculus of subgradients these algorithms rely on.

This mission is the first of a series formalizing the book. It covers §1.2 (convex functions and the concept of subgradient) and §1.3 (rules for computing subgradients), printed pages 7–16. Every later mission in the series (the subgradient method, space dilation, the ellipsoid method, decomposition) assumes that a subgradient of the objective can be computed, and §1.3 is where the book explains how: by combining subgradients of simpler pieces.

Setting

Write $E_n$ for $n$-dimensional Euclidean space with inner product $(x, y)$ and norm $\|x\|$. A function $f$ with domain a convex set $M \subseteq E_n$ is convex if its epigraph $\{(u, x) : u \ge f(x),\ x \in M\}$ is convex, equivalently $(1-\alpha) f(x_1) + \alpha f(x_2) \ge f((1-\alpha)x_1 + \alpha x_2)$ for $x_1, x_2 \in M$ and $\alpha \in [0,1]$.

Let $x_0$ be an interior point of $M$. A vector $g$ is a subgradient (or generalized gradient) of $f$ at $x_0$ if

The set of all subgradients is the subdifferential, written $G(x_0)$ or $G_f(x_0)$. For $f$ differentiable at $x_0$ it is the single gradient; for $f(x) = |x|$ on $E_1$ it is $[-1, 1]$ at $x_0 = 0$.

The one-sided directional derivative of $f$ at $x_0$ in direction $\eta$ is

A direction $\eta \ne 0$ is a direction of steepest descent at $x_0$ if $\min_{\|\xi\| = 1} f'_\xi(x_0) = f'_\eta(x_0)/\|\eta\|$.

Formalization targets

Goal: Theorem 1.12 (p. 13), the subdifferential of a nonnegative combination

For convex $f_1, \dots, f_k$ on $E_n$ and $a_1, \dots, a_k \ge 0$, the function $f = \sum_{i=1}^k a_i f_i$ is convex and, at every $x_0$,

Both inclusions are part of the goal.

Milestones

1. Theorem 1.7 (p. 9): at an interior point $x_0$ of the domain, $G(x_0)$ is nonempty, bounded, convex and closed.
2. Theorem 1.8 (p. 9): at an interior point, $f'_\eta(x_0)$ exists for every $\eta$ and $f'_\eta(x_0) = \max_{g \in G(x_0)} (g, \eta)$, with the maximum attained.
3. Corollary (p. 12): an interior point $x_0$ minimizes $f$ on $M$ if and only if $0 \in G(x_0)$.
4. Theorem 1.11 (p. 12): if $0 \notin G(x_0)$ and $g_0$ is the element of $G(x_0)$ nearest the origin, then $-g_0$ is a direction of steepest descent.
5. Theorem 1.9 (p. 11): $f$ is convex on $E_n$ if and only if $f'_\eta(x)$ exists everywhere and $t \mapsto f'_\eta(x + t\eta)$ is nondecreasing for all $x, \eta$.
6. Theorem 1.10 (p. 11): a twice continuously differentiable $f$ is convex if and only if its Hessian is positive semidefinite everywhere.
7. Theorem 1.13 (p. 14): for convex $f_1, \dots, f_m$, the function $\varphi = \max_i f_i$ is convex and $G_{f_i}(x_0) \subseteq G_\varphi(x_0)$ for every index $i$ active at $x_0$.

Significance

The result. Theorem 1.12 is the finite-dimensional, finite-valued case of the Moreau–Rockafellar sum rule. With Theorem 1.13 it is the book's recipe for computing subgradients of functions assembled from simple pieces by nonnegative combinations and pointwise maxima, the two operations that produce most nonsmooth convex objectives in practice (Lagrangian duals, penalty functions, piecewise-linear costs). Theorem 1.8 identifies the directional derivative with the support function of the subdifferential; the Corollary and Theorem 1.11 give the optimality condition and the steepest-descent direction that every descent method for nonsmooth convex functions starts from.

Formalizing it. These results are classical and have been proved many times in textbooks (Rockafellar, Convex Analysis, 1970, §23). Mathlib at the pinned revision has convexity, separation theorems and Carathéodory's theorem, but no subdifferential of a convex function on $E_n$ and no max formula. The mission builds that layer: a subdifferential with the book's inequality (1.3), a one-sided directional derivative defined as a right-hand limit, and the calculus rules above. The platform has a Clarke-gradient analogue of Theorem 1.8 for locally Lipschitz functions and a Banach-space sum rule for $f + \tfrac12\|\cdot\|^2$; neither states the convex, finite-dimensional results here.

Difficulty

The inclusion "$\supseteq$" in Theorem 1.12 is immediate from (1.3). The inclusion "$\subseteq$" is the content: a subgradient of the sum is a global object, and nothing in (1.3) splits it into subgradients of the pieces. Adding the inequalities of the pieces only produces vectors of the right form; it does not show every subgradient of $f$ arises this way. Any argument has to use the finite dimension and the interior-point setting, which is where Theorems 1.7 and 1.8 (existence of subgradients, compactness of $G(x_0)$, existence of one-sided derivatives) come in.

Theorem 1.8 in turn needs the existence of a finite right-hand limit of the difference quotient, which requires both monotonicity of the quotient and a lower bound, and the existence of a subgradient attaining the maximum, which is a separation statement. Theorem 1.9's "if" direction must rebuild convexity from one-sided derivative information alone, with no differentiability assumption.

Formalization scope

• $E_n$ is EuclideanSpace ℝ (Fin n) and $(x, y)$ is inner ℝ x y. A function is f : EuclideanSpace ℝ (Fin n) → ℝ; the domain $M$ is a set, convexity on it is ConvexOn ℝ M f (which includes convexity of $M$), and interior points are x₀ ∈ interior M. Theorems 1.9, 1.10, 1.12 and 1.13 are stated on all of $E_n$ (Set.univ), as the book proves them.
• The subdifferential subdifferential M f x₀ is the set of $g$ with $f(x) - f(x_0) \ge (g, x - x_0)$ for all $x \in M$. It is defined for any $x_0$; the interior-point assumption is a hypothesis of each theorem that needs it.
• HasOneSidedDirDeriv f x₀ η d is the right-hand limit Tendsto … (𝓝[>] 0) (𝓝 d), not Mathlib's two-sided lineDeriv. Maxima and minima are stated with IsGreatest/IsLeast (attained), never with sSup/sInf.
• Theorem 1.12 is an equality of sets over indices Fin k; $k = 0$ and zero coefficients are allowed, as in the book. Theorem 1.13's maximum is Finset.sup' over Fin m with $m \ge 1$, and it asserts only the inclusion the book states.
• Theorem 1.11 is stated as the book's proof establishes it: the steepest-descent direction is minus the minimal-norm subgradient. The printed statement names the minimal-norm subgradient itself, along which the directional derivative is positive.
• A formalization that states only "$\supseteq$" in Theorem 1.12, or only that each $\sum a_i g_i$ is a subgradient, is the easy half and does not count as the goal.
• Theorems 1.1–1.6 (supporting hyperplane, separation, representation by extremal points, the convexity inequality, continuity on the interior) are Mathlib-level (geometric_hahn_banach_*, Carathéodory and Krein–Milman, ConvexOn.continuousOn_interior) and are not restated.

Contributions welcome: proofs of any milestone, a reusable lemma that the difference quotient of a convex function is monotone in $t$, and a general max formula; these are reusable in the later missions of the series, which define subgradients the same way.

Selected references

• N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer, 1985, §§1.2–1.3, pp. 7–16. https://doi.org/10.1007/978-3-642-82118-9
• R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970, §23 (subgradients) and Theorem 23.8 (sum rule). https://doi.org/10.1515/9781400873173
• J.-J. Moreau, "Fonctionnelles sous-différentiables", Comptes Rendus de l'Académie des Sciences 257 (1963), 4117–4119.

Motivation

In a two-stage stochastic linear program a decision $x$ is taken before random data $\xi$ are observed, and a corrective recourse decision $y$ is taken afterwards at a cost. The objective contains the expectation of an optimal value of a linear program, $\int Q(x,\xi(\omega))\,P(d\omega)$, and for continuous or high-dimensional $\xi$ that integral cannot be evaluated exactly. Practical methods therefore replace $\xi$ by a discrete random vector and control the error by computable lower and upper bounds on the expected recourse cost. Chapter 2 of Ermoliev and Wets (eds.), Numerical Techniques for Stochastic Optimization (Springer 1988), by P. Kall, A. Ruszczyński and K. Frauendorfer, surveys these bounds as they were used in the codes of the time: Jensen's inequality from below, the Edmundson–Madansky inequality from above, and the special structure of simple recourse, where the expected cost is available in closed form.

Timeline. Jensen's inequality (1906) gives the lower bound for a convex integrand. A. Madansky, "Bounds on the expectation of a convex function of a multivariate random variable", Ann. Math. Statist. 30 (1959), and H. P. Edmundson (RAND report, 1956) gave the upper bound by the two-point law on the endpoints of an interval, and its product version for independent components. Kall and Stoyan (1982), Huang, Ziemba and Ben-Tal (1977), Frauendorfer and Kall (1988) developed partition refinement of both bounds, the scheme this chapter describes; Frauendorfer (1988) extended the upper bound to dependent data on boxes.

Setting

The two-stage problem (2.11) is: minimize $\psi(x)=c^Tx+\int_\Omega Q(x,\xi(\omega))\,P(d\omega)$ subject to $Ax=b$, $x\ge 0$. The recourse cost $Q(x,\xi)$ is the optimal value of the second-stage problem (2.12),

with a deterministic $m_2\times n_2$ matrix $W$ (fixed recourse), and $Q=+\infty$ when (2.12) is infeasible. Throughout the chapter the book assumes complete recourse, $\{Wy:y\ge0\}=\mathbb R^{m_2}$, and dual feasibility: for every realization of $q$ some $u$ satisfies $W^Tu\le q$. Under these assumptions $Q$ is finite. The expected recourse function is $\mathcal Q(x)=\int Q(x,\xi(\omega))\,P(d\omega)$.

The Edmundson–Madansky law of an interval $[a,b]$, $a<b$, with mean $\xi^0$ puts mass $p_1=(b-\xi^0)/(b-a)$ at $a$ and $p_2=(\xi^0-a)/(b-a)$ at $b$ (2.32). For a box $\Xi=\times_{j=1}^m[a_j,b_j]$ and means $\xi^0_j$, the vector $\hat\xi$ with independent components of these two-point laws sits at the vertex $v$ with probability $\prod_j p_j(v_j)$.

Simple recourse is the case $W=[I,-I]$, $q=[q^+,q^-]$ with $q^+_j+q^-_j\ge0$, deterministic $T$ and random $h$ only. With $\chi=Tx$ the recourse cost splits into one-row costs $Q_j(\chi_j,h_j)=q^+_j(h_j-\chi_j)$ if $h_j\ge\chi_j$, and $q^-_j(\chi_j-h_j)$ otherwise.

Formalization targets

Goal: the Edmundson–Madansky bound for independent components (p. 46)

If $\xi$ has independent components $\xi_j\in[a_j,b_j]$ with means $\xi^0_j$, and $\varphi$ is convex on $\Xi=\times_j[a_j,b_j]$, then

The book applies it to $\varphi=Q(x,\cdot)$; the goal is stated for every convex $\varphi$, with the explicit weights of (2.32).

Milestones

1. Properties (b), (d), (e) of p. 40: $Q(x,\cdot)$ is piecewise linear and convex in $(h,T)$; $Q(\cdot,\xi)$ is convex piecewise linear in $x$; the expected recourse function is finite and convex under finite second moments.
2. The Jensen lower bound (2.26)–(2.27) on a partition (a published, proved theorem, reused).
3. The dual-multiplier lower bound (2.30)–(2.31).
4. The one-dimensional Edmundson–Madansky inequality (2.32)–(2.34).
5. For simple recourse: separability (2.46)–(2.49), the closed form (2.51) of $EQ_j$, and the bounds (2.55)–(2.56) from the one-block problem.

Significance

The upper bound is the half of the bounding scheme that is not automatic. Jensen's inequality needs only a mean; an upper bound on the expectation of a convex function needs a bounded support and, in the product form, independence. Together they give a certified interval for the optimal value of a two-stage problem, and repeated partitioning of the support shrinks that interval; this is the basis of the sequential approximation methods of §2.2.4 and of later codes. The dual-multiplier bound and the simple-recourse formulas are the pieces that make those intervals cheap to compute.

The results are classical and proved in the literature cited on the page. The one-dimensional Edmundson–Madansky inequality and the general extreme-point form of the upper bound (a measure on the extreme points reproducing the barycentre) are already formalized on Prove2Me in the Introduction to Stochastic Programming series, as is the partition Jensen bound. The product form for independent components is not: deriving it from the extreme-point form requires constructing the product kernel, which is the content of this mission. The recourse properties (b), (d), (e) for a general distribution with finite second moments, the dual-multiplier bound and the simple-recourse formulas are not formalized anywhere known to this mission.

Difficulty

The obvious argument inducts on the dimension, applying the one-dimensional inequality in one coordinate while the others are held fixed. That step needs the conditional law of the remaining coordinates given the first to be their unconditional law, i.e. independence expressed as a product decomposition of the joint law, and it needs $\varphi$ with one coordinate replaced by an endpoint to remain convex on the lower-dimensional box and integrable. For dependent components the inequality is false with these weights: on $[0,1]^2$ with means $(\tfrac12,\tfrac12)$ and $\varphi(x,y)=(x-y)^2$, the product law gives $\tfrac12$ while mass $\tfrac12$ at $(1,0)$ and at $(0,1)$ gives $1$. The book's remark that the product law is extremal among all laws on $\Xi$ with the given mean fails for this reason when $m\ge2$, and is not part of this mission.

For the recourse properties the difficulty is bookkeeping: $Q$ is an extended-real optimal value, and finiteness, measurability in $\omega$ and integrability must be derived from complete recourse, dual feasibility and the moment hypothesis rather than assumed.

Formalization scope

Vectors are functions from finite index types to $\mathbb R$ (ι → ℝ), matrices are Mathlib Matrix, and random data live on a probability space (Ω, P). The recourse cost is an EReal infimum over the feasible set, so infeasibility gives $+\infty$ and unboundedness $-\infty$ exactly as on p. 39; theorems that integrate it carry complete recourse and dual feasibility, which make it finite. The expected recourse function integrates the real part of the recourse cost. Independence of the components is ProbabilityTheory.iIndepFun; the box is Set.pi univ (fun j => Icc (a j) (b j)), with $a_j<b_j$, and values in the box are required almost surely. The upper bound is the explicit sum over Boolean vertex labels of products of the weights (2.32); no abstract extremal measure is used.

Conventions fixed where the page is silent or ambiguous:

• Properties (b), (d), (e) are stated on all of $\mathbb R^{n_1}$: under the standing complete-recourse assumption $K_2=\mathbb R^{n_1}$. "Convex piecewise linear" is rendered as a maximum of finitely many affine functions.
• The book's hypothesis of finite second moments in (e) is kept as stated, componentwise.
• The book writes $Q$ for both $Q(x,\xi)$ and $\mathcal Q(x)$, and reuses $\tilde Q$, $\tilde\psi$ for different functions in (2.27) and (2.30)–(2.31); the Lean names are recourseCost, expectedRecourse and dualLowerBound.
• In (2.51) a conditional mean on a null event is $0$ in Lean; it always appears multiplied by that event's probability, so the formula is unchanged.
• In (2.56) the minimum is a real infimum over the nonempty first-stage feasible set; attainment is not claimed.
• No constant of the chapter is hidden behind $O(\cdot)$; all bounds are explicit.

A goal stated for affine $\varphi$ (where it is an equality), or with $\hat\xi$ allowed to be any discrete law with the right mean, would be trivial or a different theorem; the weights are the products of (2.32), and independence of the components is a hypothesis.

A complete development needs: finite-dimensional LP duality with extended-real values (reusable across all recourse missions), measurability and integrability of optimal-value functions, the conditional-independence step for product measures, and the one-dimensional chord inequality. The partitioned upper bound (2.37) and the discrete reformulation (2.21), (2.28) are natural follow-up statements on the same definitions.

Selected references

• P. Kall, A. Ruszczyński, K. Frauendorfer, "Approximation Techniques in Stochastic Programming", in Yu. Ermoliev and R. J-B Wets (eds.), Numerical Techniques for Stochastic Optimization, Springer Series in Computational Mathematics 10, Springer 1988, Ch. 2, pp. 33–64. https://doi.org/10.1007/978-3-642-61370-8
• A. Madansky, "Bounds on the expectation of a convex function of a multivariate random variable", Annals of Mathematical Statistics 30 (1959), 743–746. https://doi.org/10.1214/aoms/1177706203
• P. Kall, Stochastic Linear Programming, Springer 1976. https://doi.org/10.1007/978-3-642-66252-2
• R. J-B Wets, "Stochastic programs with fixed recourse: the equivalent deterministic program", SIAM Review 16 (1974), 309–339. https://doi.org/10.1137/1016053
• K. Frauendorfer, "Solving SLP recourse problems with arbitrary multivariate distributions — the dependent case", Mathematics of Operations Research 13 (1988), 377–394. https://doi.org/10.1287/moor.13.3.377
• J. R. Birge, F. Louveaux, Introduction to Stochastic Programming, Springer 1997, Ch. 8. https://doi.org/10.1007/b97617

Motivation

Algorithmic mechanism design asks for algorithms whose inputs are held by self-interested parties. Each party reports its private data, the algorithm computes an outcome, and payments are arranged so that no party gains by misreporting. Nisan and Ronen introduced the field in Algorithmic Mechanism Design (Games Econ. Behav. 35, 2001). Their running example is task scheduling on unrelated machines: $k$ tasks are distributed among $n$ machines owned by different agents, each agent knows only its own processing times, and the designer wants to minimize the make-span.

Without incentives the problem is classical: minimizing make-span on unrelated machines is NP-hard and admits a polynomial 2-approximation (Lenstra, Shmoys, Tardos, 1990). With selfish agents the question changes: which approximation ratios can a truthful mechanism guarantee? This mission formalizes the paper's upper bound, the MinWork mechanism, which is the benchmark every later lower bound for truthful scheduling is compared with.

Timeline.

• 1961: Vickrey introduces the second-price auction (J. Finance 16).
• 1971–1973: Clarke and Groves generalize it to the VCG family of truthful mechanisms for utilitarian objectives (Groves, Econometrica 41, 1973).
• 1999/2001: Nisan and Ronen show MinWork is a strongly truthful $n$-approximation, and that no truthful mechanism beats ratio 2.
• 2007: Christodoulou, Koutsoupias and Vidali raise the deterministic lower bound to $1+\sqrt2$ for $n \ge 3$; Koutsoupias and Vidali later raise it to $1+\varphi \approx 2.618$.
• 2023: Christodoulou, Koutsoupias and Kovács prove the Nisan–Ronen conjecture: no deterministic truthful mechanism achieves a ratio below $n$ (STOC 2023, arXiv:2301.11905), so MinWork is optimal among deterministic truthful mechanisms.

Setting

There are $n$ agents and $k$ tasks. Agent $i$'s type is the vector $t^i = (t^i_1,\dots,t^i_k)$ of positive times, $t^i_j > 0$ being the time agent $i$ needs to perform task $j$. A type vector is $t = (t^1,\dots,t^n)$. An allocation $x$ sends each task $j$ to one agent; $x^i$ is the set of tasks agent $i$ receives. The make-span of $x$ is

and agent $i$'s valuation is $v^i(x,t^i) = -\sum_{j \in x^i} t^i_j$.

A direct mechanism asks every agent to declare a type, computes an allocation $x(d)$ from the declared vector $d$, and hands agent $i$ a payment $p^i(d)$. Agent $i$'s utility is $p^i(d) + v^i(x(d), t^i)$, with $t^i$ its true type. The mechanism is truthful if declaring $t^i$ maximizes agent $i$'s utility for every declaration of the others, and strongly truthful if truth-telling is the only such dominant strategy. An allocation rule is a $c$-approximation if $g(x(t),t) \le c \cdot g(y,t)$ for every type vector $t$ and every allocation $y$.

The MinWork mechanism allocates each task to an agent with minimal declared time for it, breaking ties arbitrarily. For each task it wins, an agent receives the second-best declared time $\min_{i' \ne i} d^{i'}_j$:

The Lean development uses the same names: load, makespan, IsTruthful, IsStronglyTruthful, IsApprox, IsMinWorkAlloc, secondBest, minTime, minWorkPay.

Formalization targets

Goal: Theorem 4.1

For $n \ge 2$ and every MinWork allocation rule $x$ with payments $p$ as above,

Milestones

1. Theorem 3.1 (Groves): a VGC mechanism is truthful. This is an existing platform theorem, used as a reference.
2. MinWork belongs to the VGC family. Its allocation maximizes $\sum_i v^i(t^i,x)$, and its payment is $\sum_{i'\ne i} v^{i'}(t^{i'},x(t)) + h^{-i}$ with $h^{-i} = \sum_j \min_{i'\ne i} t^{i'}_j$.
3. Claim 4.2: MinWork is strongly truthful.
4. $g(x(t),t) \le \sum_{j} \min_i t^i_j$.
5. $g(y,t) \ge \frac1n \sum_j \min_i t^i_j$ for every allocation $y$.
6. Claim 4.3: MinWork is an $n$-approximation.

Significance

The theorem gives the first positive result for truthful scheduling: a mechanism that is truthful in the strongest sense and is within a factor $n$ of optimal, whatever the tie-breaking rule. Every lower bound in the paper (Theorems 4.6, 4.10 and 4.12) and in the later literature measures itself against this ratio. Since the 2023 resolution of the Nisan–Ronen conjecture, the ratio $n$ is known to be tight for deterministic truthful mechanisms.

The result is proved in the paper; it is not known to be formalized in any proof assistant. The platform already has Groves' theorem in an abstract form (AGT.vcg_incentive_compatible). This mission connects that abstract statement to a concrete combinatorial mechanism, and it adds the strict part of strong truthfulness for any number of tasks and agents, which the paper proves only for one task and two agents. The vocabulary (make-span over unrelated machines, direct scheduling mechanisms, strong truthfulness) is shared with the seven later missions of this series.

Difficulty

Truthfulness follows from Groves' theorem once MinWork is identified as a VGC mechanism. The identification requires the payment identity at every declared vector and under every tie-breaking rule, including ties at the winning time. The main difficulty is the strict part of strong truthfulness. A misreport that differs from the truth only on one task must still be shown to lose strictly for some declarations of the others. Those declarations must stay positive, and on every other task they must leave the outcome unchanged. The paper's proof covers only one task and two agents and leaves the general case as "similar". Its printed inequality also has the two utilities in the wrong order (see below), so it cannot be transcribed directly.

Formalization scope

• Agents are Fin n and tasks are Fin k. An allocation is a function Fin k → Fin n, and an agent may receive no task. Types are positive reals, and every truthfulness and approximation quantifier ranges over positive true types, positive misreports and positive declarations of the others.
• Payments are handed to the agent, so utility is the payment minus the true time spent. Payments are computed from the declared vector, never from true types.
• The allocation rule is a parameter satisfying the MinWork specification (IsMinWorkAlloc). Every result holds for every tie-breaking rule, including rules that depend on the whole declared vector. No particular argmin is fixed.
• $n \ge 2$ is a hypothesis of the goal and of the truthfulness items: with a single agent the paper's second-best minimum is undefined. The approximation items need only $n \ge 1$. There is no hypothesis on $k$.
• The make-span and both minima are Finset.sup' / Finset.inf' over nonempty finite sets, so they are true maxima and minima with no default values.
• Strong truthfulness is formalized as truthfulness plus: every misreport $d^i \ne t^i$ is strictly worse than the truth for some positive declarations of the others. Given truthfulness this is equivalent to Definition 5. A formalization that states only that truth-telling is dominant, or proves strictness only for single-task instances, does not meet the goal. Neither does an existential ratio in place of $n$.
• Printed slip: in the proof of Claim 4.2 (p. 177) the case $d^i > t^i$ reads "the utility for agent $i$ is $t^i - d^i < 0$, instead of 0 in the case of truth-telling". With the Definition 11 payments the misreporting agent loses the task (utility 0), and the truthful agent wins it with utility $d^{3-i} - t^i > 0$. The milestone text keeps the paper's words; the Lean statements assert what the argument establishes.
• Out of scope: running time ("polynomial time"), and the paper's general revelation-principle framework (Proposition 2.1).
• Welcome contributions: proofs of the milestones, and a reusable lemma connecting the local VGC milestone to AGT.vcg_incentive_compatible.

Selected references

• N. Nisan, A. Ronen, Algorithmic Mechanism Design, Games and Economic Behavior 35 (2001) 166–196. https://doi.org/10.1006/game.1999.0790
• T. Groves, Incentives in Teams, Econometrica 41 (1973) 617–631. https://doi.org/10.2307/1914085
• W. Vickrey, Counterspeculation, Auctions, and Competitive Sealed Tenders, Journal of Finance 16 (1961) 8–37. https://doi.org/10.1111/j.1540-6261.1961.tb02789.x
• J. K. Lenstra, D. B. Shmoys, É. Tardos, Approximation algorithms for scheduling unrelated parallel machines, Mathematical Programming 46 (1990) 259–271. https://doi.org/10.1007/BF01585745
• G. Christodoulou, E. Koutsoupias, A. Kovács, A Proof of the Nisan-Ronen Conjecture, STOC 2023. https://arxiv.org/abs/2301.11905

Motivation

Column generation solves linear programs with far too many variables to list: one works with a small subset $J' \subseteq J$ of the columns, the restricted master problem (RMP), and adds columns of negative reduced cost as a pricing problem finds them (Lübbecke and Desrosiers 2005, §2.1). Many integer programs from vehicle routing, crew scheduling and crew pairing are reformulated so that the master problem is a set-partitioning problem: every row (a customer, a flight leg, a task) must be covered by exactly one selected column (a route, a pairing, a schedule). The linear relaxation of such a master is solved by column generation, and an integer solution is then sought by branch-and-price, branch-and-bound with column generation at every node.

Branching in branch-and-price is not free. Fixing a master variable $\lambda_j$ to $0$ does not stop the pricing problem from regenerating the same column, and handling that complicates the pricing problem. The rule that avoids this for set partitioning goes back to Ryan and Foster (1981): branch on a pair of rows, requiring them to be covered either by the same column or by two different columns. Both requirements are constraints the pricing problem can respect directly. Lübbecke and Desrosiers call it "the most common scheme in conjunction with column generation" (§7.3, p. 1020) and state the proposition that makes it well defined as their Proposition 3.

Timeline:

• 1981: Ryan and Foster introduce the pair-of-rows branching rule for set partitioning in crew scheduling.
• 1998: the branch-and-price survey of Barnhart et al. (1998) presents the rule as the branching scheme for set-partitioning masters.
• 2005: Lübbecke and Desrosiers state it as Proposition 3 of their survey, attributing it to Ryan and Foster without proof.

Setting

Rows are indexed by $\{1, \dots, m\}$ and columns by a finite set $J'$. A matrix $A = (a_{rj}) \in \{0,1\}^{m \times |J'|}$ has every entry equal to $0$ or $1$; column $j$ covers row $r$ when $a_{rj} = 1$. The set-partitioning system of the RMP's linear relaxation is

A vector $\lambda$ is a basic feasible solution of this system when it satisfies all its constraints and, among the constraints active at $\lambda$ (the $m$ equality rows, and the constraints $\lambda_j \ge 0$ with $\lambda_j = 0$), there are $|J'|$ linearly independent ones. Equivalently, $\lambda$ is feasible and the columns of $A$ in the support of $\lambda$ are linearly independent. A solution is fractional when it is not a $0/1$ vector, $\lambda \notin \{0,1\}^{|J'|}$.

For two rows $r, s$ the Ryan–Foster quantity is $\sum_{j \in J'} a_{rj} a_{sj} \lambda_j$, written pairCover A lam r s in Lean: the total weight on the columns that cover both rows. For $r = s$ it is the row sum, equal to $1$ on every feasible $\lambda$.

Formalization targets

Goal: Proposition 3 (p. 1020)

For every $0/1$ matrix $A$ and every fractional basic feasible solution $\lambda$ of $A\lambda = \mathbf 1$, $\lambda \ge \mathbf 0$,

The two rows are automatically distinct. The statement concerns every fractional basic solution, not only an optimal one, and no cost vector enters it.

Milestone: the branches keep every integer solution (§7.3, p. 1020)

For every $0/1$ solution $\lambda \in \{0,1\}^{|J'|}$ of $A\lambda = \mathbf 1$ and every pair of rows $r, s$,

This is the paper's requirement that "integer solutions remain intact" (p. 1019), specialised to the two branches "$= 1$" and "$= 0$" of the paragraph after Proposition 3.

Significance

Proposition 3 is what makes Ryan–Foster branching a valid branching scheme in the sense of §7.3: the current fractional solution violates both branches for the chosen pair, so it is excluded from both children, while by the milestone every integer solution survives in one of them. The same pair-of-rows idea underlies branching in bin packing, graph colouring, vehicle routing and crew scheduling codes, where it is used because both branches translate into constraints on the pricing problem rather than on individual master variables.

The paper states the result and refers its proof to Ryan and Foster (1981); no machine-checked version of the proposition is known. This mission produces a formal statement tied to a standard, representation-aware definition of basic solutions (Bertsimas–Tsitsiklis Definition 2.9, already on the platform) and, once proved, a verified lemma that any formal development of branch-and-price for set partitioning can cite.

Difficulty

An argument that uses only feasibility and a fractional coordinate cannot work. Without basicness the claim is false: with one row and two identical columns, $A = [1\ 1]$, the vector $\lambda = (\tfrac12, \tfrac12)$ is feasible and fractional, yet the only pair of rows is $r = s$, whose quantity is $1$. The difficulty is to turn basicness, a linear-algebra condition, into a combinatorial statement about which rows the fractional columns cover. A set-partitioning matrix need not have full row rank, so the familiar description of basic solutions through an invertible basis matrix is not available in general.

Formalization scope

• Rows are Fin m and columns Fin n, so $J'$ is identified with $\{0, \dots, n-1\}$. $A$ is a real matrix Matrix (Fin m) (Fin n) ℝ with the hypothesis IsZeroOneMatrix A (every entry $0$ or $1$); $\lambda$ is lam : Fin n → ℝ, since λ is a Lean keyword. Columns are $0/1$ vectors; the equivalent reading as subsets of the rows is only prose.
• "Basic solution" is not defined in the paper. It is read as Bertsimas–Tsitsiklis Definition 2.9 for the standard-form constraint family: LinearOptimization.IsBasicFeasibleSolution (LinearOptimization.stdFormSystem A (fun _ => 1)) lam, from the platform definitions BasicSolution and ActiveConstraints. This definition needs no full-row-rank assumption, which a set-partitioning matrix need not satisfy, and it is not replaced by an ad hoc support condition.
• Typo correction. The paper writes "i.e., $\lambda \notin \{0,1\}^m$". Since $\lambda$ has one coordinate per column, the statement reads it as $\lambda \notin \{0,1\}^{|J'|}$: ¬ IsZeroOneVector lam.
• The rows $r, s$ range over all of $\{1, \dots, m\}$, including $r = s$, as on the page; no distinctness is assumed or required.
• "Fractional basic solution" is any such solution, not the RMP optimum; no costs or optimality hypothesis enter.
• The milestone reads the paragraph after Proposition 3, together with the validity requirement "integer solutions remain intact" (p. 1019), as the dichotomy for all $0/1$ solutions of $A\lambda = \mathbf 1$. The sentence about transferring the branching information to the pricing problem is not formalized.
• Edge cases: for $m = 0$ basicness forces $\lambda = 0$, and for $n = 0$ the vector is empty; in both cases no fractional basic solution exists and the goal is vacuous, as on the page.
• A trivializing formalization is ruled out: dropping basicness makes the goal false (the $[1\ 1]$ example above), dropping the $0/1$ hypothesis on $A$ changes the meaning of the quantity, and replacing "fractional basic" by an unsatisfiable hypothesis would make it empty; the hypotheses are satisfied, for instance, by three rows, the columns $\{1,2\}, \{2,3\}, \{1,3\}$ and $\lambda = (\tfrac12, \tfrac12, \tfrac12)$.

A complete development needs linear-algebra facts about basic solutions of standard-form systems without a rank assumption (support columns linearly independent), which are reusable beyond this mission. Contributions welcome: that characterization as a lemma, and proofs of the milestone and of the goal.

Selected references

• M. E. Lübbecke and J. Desrosiers, Selected Topics in Column Generation, Operations Research 53(6):1007–1023, 2005. https://doi.org/10.1287/opre.1050.0234
• D. M. Ryan and B. A. Foster, An integer programming approach to scheduling, in A. Wren (ed.), Computer Scheduling of Public Transport, North-Holland, 1981, pp. 269–280.
• C. Barnhart, E. L. Johnson, G. L. Nemhauser, M. W. P. Savelsbergh and P. H. Vance, Branch-and-Price: Column Generation for Solving Huge Integer Programs, Operations Research 46(3):316–329, 1998. https://doi.org/10.1287/opre.46.3.316
• D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, 1997, Definition 2.9.

Motivation

Many optimization problems in engineering, simulation-based design and machine learning give access only to function values: the objective is the output of a simulator or a black-box program, and its gradient is unavailable or too expensive. Derivative-free (or zeroth-order) methods address this setting. Nesterov and Spokoiny (Found. Comput. Math. 17 (2017)) showed that a very simple oracle, the finite difference of $f$ along a random Gaussian direction, can replace the gradient in standard first-order schemes at the price of a factor depending only on the dimension. Their analysis became the reference point for later work on zeroth-order stochastic optimization and on gradient-free methods in reinforcement learning and adversarial attacks.

This mission covers Section 6 of the paper: the accelerated random method $\mathcal{FG}_\mu$ and its rate, Theorem 9. It is the third mission of a series; the first covers random search for nonsmooth problems (Theorem 6), the second the random gradient method for smooth problems (Theorem 8).