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

$$R_S = f_2 \min(r_2, C - S) + f_1 \min\bigl(r_1,\, C - \min(r_2, C - S)\bigr).$$

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

$$\mathrm{EMSR}_1(S) = f_1 \cdot \bar P_1(S).$$

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

$$\mathrm{EMSR}_1(S) \ge f_2 .$$

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

$$\bar R(S) \le \bar R(S_2^1) \qquad \text{for all } S \in \{0, \dots, C\}.$$

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

Why airlines allocate seats by fare class

An airline sells the seats of one flight leg at several prices. Low fares fill seats that would otherwise fly empty; high fares are bought by passengers who book late and cannot be predicted exactly. Seat inventory control decides how many seats each fare class may sell. Peter Belobaba's 1987 MIT dissertation (MIT Flight Transportation Laboratory Report R87-7) gave the probabilistic treatment of this problem that became the expected marginal seat revenue (EMSR) method, which was in use across the airline industry for decades.

This mission formalizes the first, simplest model of the thesis: distinct (non-nested) fare-class inventories on a single leg, where a seat assigned to a class may be sold only in that class or not at all. The thesis surveys this model in Sect. 4.2 (pp. 84–94), where an integer-programming formulation from McDonnell-Douglas and its solution by ranking marginal values are described (p. 90), and develops it probabilistically in Sect. 5.1 (pp. 102–107).

Setting

A leg has capacity $n$ seats (the thesis also writes $C$). There are finitely many fare classes $i$. Class $i$ has an average fare $f_i \ge 0$ and receives a random number of requests $r_i \in \{0,1,2,\dots\}$, with law $p_i$. The seats are split into allocations $S_i \in \mathbb{N}$, one per class.

With $S$ seats, a class books requests until its seats run out, so its bookings and spill (refused requests) are

$$b = \min(r, S), \qquad l = (r - S)^+ \qquad \text{(Eq. (5.3))}.$$

The expected revenue of class $i$ is $\bar R_i(S_i) = f_i \cdot \bar b_i(S_i)$ with $\bar b_i(S_i) = E[\min(r_i,S_i)]$, and the leg's expected revenue is $\bar R = \sum_i \bar R_i(S_i)$ (Eq. (5.9)). Write

$$\bar P_i(S) = P[r_i \ge S], \qquad \mathrm{EMSR}_i(S) = f_i \cdot \bar P_i(S) \qquad \text{(Eqs. (5.11), (6.1), (6.2))},$$

the expected marginal seat revenue of the $S$-th seat of class $i$. The value of the $k$-th seat of class $i$ in the integer program of p. 90 is $m_i(k) = \mathrm{EMSR}_i(k)$, $k = 1, \dots, n$.

Formalization targets

Goal: the $n$ largest marginal values give the optimal booking limits (p. 90)

Let $T$ be any set of $n$ pairs $(i,k)$, $1 \le k \le n$, such that every $m_i(k)$ with $(i,k) \in T$ is at least every $m_j(l)$ with $(j,l) \notin T$, and let $S^T_i = \#\{k : (i,k) \in T\}$. Then $\sum_i S^T_i = n$ and

$$\sum_i f_i\, E[\min(r_i, S_i)] \;\le\; \sum_i f_i\, E[\min(r_i, S^T_i)] \qquad \text{for every } S \text{ with } \textstyle\sum_i S_i \le n .$$

The goal fixes no distribution, number of classes or fare ordering, and it holds for every tie-breaking among equal marginal values.

Milestones

1. Eq. (5.6): $\bar b_i(S) + \bar l_i(S) = \bar r_i$ for requests of finite mean.
2. Eq. (5.11): $\bar R_i(S) - \bar R_i(S-1) = f_i \cdot P[r_i \ge S]$ for $S \ge 1$.
3. Eqs. (6.1)–(6.2): $\bar P_i$ and $\mathrm{EMSR}_i$ are non-increasing in $S$.
4. Eq. (4.5): the 0–1 vector equal to $1$ on a set of $n$ largest $m_i(k)$ is an optimal solution of the linear program $\max \sum_{i,k} X_{ik} m_i(k)$ subject to $\sum X_{ik} \le n$, $0 \le X_{ik} \le 1$.
5. Eq. (5.13), discrete form: an allocation of exactly $C$ seats maximises $\bar R$ among such allocations if and only if some $\lambda$ satisfies $\mathrm{EMSR}_i(S_i) \ge \lambda$ whenever $S_i \ge 1$ and $\mathrm{EMSR}_i(S_i + 1) \le \lambda$, for all $i$.

Significance

The goal is the reason distinct-inventory allocation is computationally easy: a revenue-maximising allocation is obtained by sorting $n \times (\text{number of classes})$ numbers, with no search over allocations. The same marginal-value principle underlies the EMSR rules for nested classes in the rest of the thesis, and the identity (5.11) is the link between an expected-revenue function and its marginal seat values used throughout revenue management. Milestone 5 is the integer form of the Lagrangian condition of Eq. (5.13); the thesis states it only for a continuous relaxation, and its equality form is generally unattainable with integer seats.

These results are classical and their proofs are elementary; to our knowledge none of them has been machine-checked. The mission produces a reusable formal model of a single-leg, distinct-inventory allocation problem with integer demand (bookings, spill, revenue and marginal values of a PMF ℕ), and checked statements of the marginal-allocation principle for it.

Difficulty

The thesis argues with continuous densities and derivatives, setting $\partial \bar R / \partial S_i$ equal across classes. That argument does not transfer to integer seats: the derivative of a step-shaped expected-revenue function does not exist, equality of marginal values across classes generally fails at every integer allocation, and the tail probability must be $P[r \ge S]$ rather than the $P[r > S]$ of Eq. (5.2) for the marginal identity to hold. The integer statements need their own exchange argument. A second subtlety is ties: "the $n$ largest values" is not unique, and the goal must hold for every admissible choice, including choices in which a class's selected seat numbers are not an initial segment $\{1, \dots, S_i\}$.

Formalization scope

All declarations live in the namespace SeatInventory.Distinct. Conventions:

• Integer demand. The law of class $i$'s requests is d i : PMF ℕ; expectations are series over $\mathbb{N}$. The thesis's continuous densities are replaced by this discrete model, which the thesis itself requires for seat allocations (p. 103).
• Tail convention. $\bar P(S) = P[r \ge S]$, as in Eq. (6.2) and the prose of Eq. (5.11) ("the probability of selling $S_i$ or more seats"), not the $P[r > S]$ of Eq. (5.2).
• Seat numbers start at 1, and the pairs $(i,k)$ range over $k \in \{1, \dots, n\}$, as the 600 variables of a 150-seat, four-class problem on p. 90 indicate.
• Nonnegative fares $f_i \ge 0$ are assumed in every statement that needs them; with a negative fare the capacity constraint $\sum S_i \le n$ would not bind and the claims fail.
• Finite mean of the requests is assumed explicitly for Eq. (5.6); the thesis assumes it silently. Expected bookings are bounded and need no assumption.
• Capacity. The goal and the LP compare against allocations with $\sum_i S_i \le n$ (the LP's constraint); milestone 5 compares allocations of exactly $C$ seats (Eq. (5.8)).
• No independence assumption. Expected revenue of distinct inventories depends only on each class's marginal law, so the statements take one law per class.
• "Decreasing" is non-increasing. The thesis's justification in Sect. 6.1.1 gives only monotonicity; strict decrease fails for bounded demand.
• LP integrality. "The solution will be integer" is stated as: the indicator of every set of $n$ largest values is optimal. With ties the LP also has fractional optima.

The expected revenue in the goal is computed from the booking rule $\min(r_i, S_i)$; it is not defined as a sum of marginal values, and the optimal allocation is not defined as an argmax of $\bar R$. Either shortcut would make the goal a tautology and is ruled out.

Needed infrastructure: tail sums of a PMF ℕ, telescoping of $E[\min(r, S)]$, and a finite exchange argument for sums of the $n$ largest values of a function on a finite set; the last two are reusable for any separable concave resource-allocation problem. Contributions of proofs of any milestone, and of a verified sorting routine that produces a set of $n$ largest values, 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).
• P. P. Belobaba, Airline yield management: an overview of seat inventory control, Transportation Science 21(2), 63–73, 1987. https://doi.org/10.1287/trsc.21.2.63
• P. P. Belobaba, Application of a probabilistic decision model to airline seat inventory control, Operations Research 37(2), 183–197, 1989. https://doi.org/10.1287/opre.37.2.183
• K. Littlewood, Forecasting and control of passenger bookings, AGIFORS Symposium Proceedings 12, 1972; reprinted in Journal of Revenue and Pricing Management 4(2), 111–123, 2005. https://doi.org/10.1057/palgrave.rpm.5170134

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:

$$\text{maximize } \sum_{j=1}^n c_j x_j \quad\text{subject to}\quad \sum_{j=1}^n a_{ij}x_j \le b_i\ (i=1,\dots,m),\qquad x_j \ge 0\ (j=1,\dots,n).$$

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

$$\zeta = \bar\zeta + \sum_{j\in\mathcal N}\bar c_j x_j,\qquad x_i = \bar b_i - \sum_{j\in\mathcal N}\bar a_{ij}x_j\quad (i\in\mathcal B).$$

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

$$D_0\to D_1\to D_2\to\cdots$$

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

The subgradient method for a nonsmooth convex function converges, but slowly: when the level sets of the objective are elongated, the subgradient is nearly orthogonal to the direction towards the minimum, and the method zigzags. For smooth functions the remedy is a change of metric (Newton and quasi-Newton methods); for nonsmooth functions no Hessian exists to supply one. N. Z. Shor's answer was to learn a metric from the subgradients themselves: after each step, stretch the space along the latest (transformed) subgradient, so that components of future subgradients parallel to it are damped. These subgradient methods with space dilation along the gradient (SDG methods) are the ancestors of Shor's r-algorithm and of the ellipsoid method, which the book (p. 49) describes as a special case of the same family and which Khachiyan later used to show that linear programming is solvable in polynomial time.

Section 3.4 of Shor's monograph (Springer 1985, translated by K. C. Kiwiel and A. Ruszczyński, doi:10.1007/978-3-642-82118-9) proves that, under a two-sided condition on the objective, a suitable SDG method decreases function values at the speed of a geometric progression whose ratio is invariant under nonsingular linear changes of variables. This mission formalizes that chain of results.

Setting

Let $E_n$ be the $n$-dimensional Euclidean space with inner product $(x,y)$. For a unit vector $\xi$ and a coefficient $\alpha \ge 0$, the operator of space dilation along $\xi$ is

$$R_\alpha(\xi)\,x = x + (\alpha - 1)(x,\xi)\,\xi ,$$

which multiplies the component of $x$ along $\xi$ by $\alpha$ and leaves the orthogonal complement fixed.

Let $f : E_n \to \mathbb{R}$ and let $g : E_n \to E_n$ be a generalized gradient: a subgradient of $f$ when $f$ is convex, an almost-gradient when $f$ is almost differentiable. The SDG method starts from $x_0$ and a nonsingular operator $B_0 = A_0^{-1}$. At step $k = 0, 1, \dots$: if $g(x_k) = 0$ it stops; otherwise it forms the transformed gradient $\tilde g_k = B_k^* g(x_k)$, the direction $\xi_{k+1} = \tilde g_k/\|\tilde g_k\|$, and

$$x_{k+1} = x_k - h_{k+1} B_k \xi_{k+1}, \qquad B_{k+1} = B_k R_{1/\alpha_{k+1}}(\xi_{k+1}), \qquad A_{k+1} = R_{\alpha_{k+1}}(\xi_{k+1}) A_k ,$$

with a stepsize $h_{k+1}$ and a dilation coefficient $\alpha_{k+1}$. So $A_k$ is the accumulated space transformation, $B_k = A_k^{-1}$, and each step is a subgradient step for $\varphi_k(y) = f(B_k y)$ in the variables $y = A_k x$.

The quantitative results assume, for a point $x^*$ and the ball $S_d = \{x : \|x - x^*\| \le d\}$, the two-sided condition

$$N\,[f(x) - f(x^*)] \le (g(x),\, x - x^*) \le M\,[f(x) - f(x^*)], \qquad x \in S_d, \quad M > N > 0. \qquad (3.18)$$

For a convex function the lower inequality holds with $N = 1$; the upper one bounds how far $f$ is from a positively homogeneous function around $x^*$.

Formalization targets

Goal: Theorem 3.4

Under (3.18), with $B_0 = I$, $x_0 \in S_d$, stepsizes $h_{k+1} = \frac{2MN}{M+N}\frac{f(x_k)-f(x^*)}{\|\tilde g_k\|}$, a constant coefficient $1 < \alpha \le \frac{M+N}{M-N}$, and $G$ a bound for $\|g\|$ on $S_d$: there are $c > 0$ and indices $k_1 < k_2 < \cdots$ with

$$f(x_{k_p}) - f(x^*) \le c\,\alpha^{-k_p/n},$$

and for every $k \ge 1$

$$\min_{0 \le i \le k-1}\,[f(x_i) - f(x^*)] \le \frac{G\sqrt{k(\alpha^2-1)}\,d}{N\sqrt{\alpha^{2k/n}-1}} .$$

Milestones

1. Eq. (3.4): $\|R_\alpha(\xi)x\| = \sqrt{\|x\|^2 + (\alpha^2-1)(x,\xi)^2}$.
2. Theorem 3.1: if $\|g(x_k)\| \le d$ and $1+\delta \le \alpha_k \le \alpha^*$, then $\|\tilde g_{k_p}\| < c\,(\prod_{j\le k_p}\alpha_j)^{-1/n}$ along a subsequence.
3. Theorem 3.2: for constant $\alpha > 1$ and $B_0 = I$, $\min_{0\le r\le k-1}\|\tilde g_r\| \le d\sqrt{k(\alpha^2-1)}/\sqrt{\alpha^{2k/n}-1}$.
4. Theorem 3.3: under (3.18) and the rules above, $\|A_k(x_k - x^*)\| \le d$ for all $k$.

Significance

The result shows that one fixed rule, depending only on $M$, $N$ and $n$, yields linear convergence of function values for every objective satisfying (3.18), at a ratio $\alpha^{-1/n}$ that does not deteriorate when the problem is badly scaled: the method, and hence its rate, is invariant under nonsingular linear changes of variables. This is the property the ellipsoid method inherits (Section 3.8 of the book), and the same space-dilation machinery drives the r-algorithm (Section 3.7), still used for large nonsmooth problems such as Lagrangian duals of integer programs.

The theorems are proved in the book; none of them, and no space-dilation method, has a machine-checked proof to our knowledge, and the platform has no statement about variable-metric subgradient methods. The mission provides a reusable formal model of the SDG iteration, the eigenvalue-growth arguments behind Theorems 3.1–3.2, and the one-step invariant of Theorem 3.3. The formalization also corrects two points of the printed text (see Formalization scope).

Difficulty

Theorem 3.3 is a one-step computation, but Theorems 3.1 and 3.2 are not: they relate the size of the transformed gradients to the growth of the singular values of $A_k$, whose determinant is $\prod_j \alpha_j$. A bound on $\|\tilde g_k\|$ at a single step says nothing, since the dilations can concentrate in few directions; the argument has to control the largest singular value of $A_k$ over many steps against the geometric-mean lower bound $(\det A_k)^{1/n}$. The dimension $n$ enters the rate exactly through this comparison. Theorem 3.4 then needs the invariant of Theorem 3.3 to keep every iterate inside $S_d$, where (3.18) and the bound $G$ are available.

Formalization scope

• $E_n$ is EuclideanSpace ℝ (Fin n) with $n \ge 1$ in the rate statements. Operators are continuous linear maps; $B_0$ is a continuous linear equivalence and $A_0$ its inverse. The state $(x_k, B_k, A_k)$ is produced by a defined recursion sdg, not assumed; $\tilde g_k$ is gTilde.
• The stepsize rule receives the index, the current point and $\tilde g_k$; the dilation coefficient at step $k$ is α (k+1). When $g(x_k) = 0$ the state is repeated, which encodes the book's stop; no division by zero is used.
• The book states Theorems 3.1–3.4 for almost differentiable $f$ with $g$ an almost-gradient; the proofs use only the bounds on $\|g\|$ and (3.18), so the Lean statements quantify over every map $g$ with those properties. $G$ is any bound for $\|g\|$ on $S_d$ in place of the maximum.
• Theorems 3.2–3.4 take $B_0 = I$, as their proofs do; Theorem 3.1 allows any nonsingular $B_0$.
• Two corrections to the printed statements, both following the proofs: Theorem 3.4's record bound carries the factor $1/N$ that the proof derives, and the record minima in Theorems 3.2 and 3.4 range over the indices $0, \dots, k-1$ that the proof controls, rather than $1, \dots, k$.
• Constants $c$ and subsequences are existential and chosen after the data of the run, before the index $p$. A statement placing $c$ after $p$, or dropping (3.18) on $S_d$, would be trivially true or false and is ruled out.

Useful infrastructure, reusable for the r-algorithm and ellipsoid chapters: identities for $R_\alpha(\xi)$, determinants and singular values of products of rank-one dilations, and the invariance of the SDG iteration under a linear change of variables. Proofs of any milestone, and alternative arguments for Theorem 3.1, are welcome.

Selected references

• N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer, 1985, §§3.2–3.4, pp. 49–62 (translated by K. C. Kiwiel and A. Ruszczyński). https://doi.org/10.1007/978-3-642-82118-9

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

$$f(x) - f(x_0) \ge (g, x - x_0) \qquad \text{for all } x \in M. \tag{1.3}$$

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

$$f'_\eta(x_0) = \lim_{t \to 0+} \frac{f(x_0 + t\eta) - f(x_0)}{t}.$$

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 $x_0$,

$$G_f(x_0) = \Big\{ \sum_{i=1}^k a_i g_i \;:\; g_i \in G_{f_i}(x_0),\ i = 1, \dots, k \Big\}.$$

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

Many engineering and economic planning problems must meet random requirements with a prescribed reliability: a reservoir must satisfy demand with probability at least $0.95$, a power system must cover load except on rare days, an inventory must avoid shortage with high probability. Probabilistic constrained programming (also called chance-constrained programming) models this by requiring that a system of random inequalities hold jointly with probability at least $p$.

The first obstacle to solving such problems is structural. The probability that random constraints are satisfied is, in general, neither concave nor convex in the decision, so the feasible set need not be convex and local search can stall. Prékopa's theory of logarithmically concave measures (1971–1973) removed that obstacle for a large class of distributions, and it is the basis of the numerical methods of Chapter 5 of Ermoliev and Wets, Numerical Techniques for Stochastic Optimization (Springer 1988). This mission formalizes the structural theorems of that chapter.

Timeline:

• 1959: Charnes and Cooper, individual chance constraints.
• 1971: Prékopa, logarithmic concave measures with application to stochastic programming (Acta Sci. Math. Szeged 32).
• 1973: Prékopa, logarithmic concave measures and functions (Acta Sci. Math. Szeged 34), containing the marginal theorem: marginals of log-concave functions are log-concave.
• 1970s–1980s: nonlinear programming methods for (5.1) (SUMT with logarithmic penalty, supporting hyperplanes, reduced gradients) combined with Monte Carlo evaluation of $h_0$; the chapter surveys them.

Setting

Let $\xi$ be a random vector in $\mathbb R^q$ on a probability space $(\Omega, \mathcal F, P)$ and let $g_1, \dots, g_r : \mathbb R^n \times \mathbb R^q \to \mathbb R$. The chapter studies problem (5.1):

$$\min h(x) \quad \text{s.t.} \quad h_0(x) = P\bigl(g_1(x,\xi) \ge 0, \dots, g_r(x,\xi) \ge 0\bigr) \ge p, \quad h_1(x) \ge p_1, \dots, h_m(x) \ge p_m .$$

The function $h_0$ is the probability function (chanceProb in Lean). In the special case $g_i(x,y) = T_i x - y_i$ it equals $F(Tx)$, where $F$ is the joint distribution function of $\xi$.

A function $f \ge 0$ is logarithmically concave on a convex set $S$ if

$$f(\lambda u + (1-\lambda) v) \ge f(u)^{\lambda} f(v)^{1-\lambda}, \qquad u, v \in S,\ 0 < \lambda < 1 .$$

Where $f > 0$ this is concavity of $\log f$; the power form also makes sense where $f = 0$. Nondegenerate normal densities, uniform densities on convex bodies and exponential densities are log-concave.

Section 5.7 introduces the polynomial distribution (5.19) on the cube $0 < z_j \le 1$:

$$F(z_1, \dots, z_n) = \frac{1}{\sum_{i=1}^{N} c_i z_1^{\alpha_{i1}} \cdots z_n^{\alpha_{in}}}, \qquad c_i > 0,\ \alpha_{ij} \le 0,\ \textstyle\sum_j \alpha_{ij} < 0$$

(polyDistF in Lean).

Formalization targets

Goal: Theorem 5.1

If $g_1, \dots, g_r$ are jointly concave on $\mathbb R^{n+q}$ and $\xi$ has a log-concave density $f$ on $\mathbb R^q$, then

$$h_0 \text{ is logarithmically concave on } \mathbb R^n .$$

Its immediate consequence is that the feasible set $\{x : h_0(x) \ge p\}$ is convex for every $p$.

Milestones

1. Theorem 5.2.1: if $h$ is log-concave on the convex set $H = \{h \ge p\}$, $0 < p < 1$, then $h - p$ is log-concave on $H$. This makes the logarithmic penalty function (5.5) of the SUMT method convex.
2. Theorem 5.2.2: under the standing assumptions of §5.2, every interior point $z$ of the feasible set of (5.1) satisfies $h_i(z) > p_i$, $i = 0, \dots, m$.
3. Theorem 5.7.1 (proved content, (5.21)): for $n = 2$ and oppositely ordered exponents, $\partial^2 F / \partial z_1 \partial z_2 \ge 0$ on $(0,1)^2$.
4. Theorem 5.7.2: the polynomial distribution function is log-concave on $(0,1]^n$.

The platform theorem ConvexOptimization.prekopa_marginal_log_concave (Prékopa's marginal theorem, proved) is included as a reference item.

Significance

Theorem 5.1 turns a probabilistic constraint into a convex constraint after taking logarithms. This is what makes convergence proofs for nonlinear programming methods (SUMT with logarithmic penalty, supporting hyperplanes, reduced gradients) applicable to (5.1) and to the reliability maximization problem (5.4); without it, those methods have no guarantee of finding a global optimum. Theorems 5.2.1 and 5.2.2 are the two facts that make the SUMT method of §5.2 well defined and convex on the interior of the feasible set. Theorem 5.7.2 shows that probabilistic constraints under the polynomial distribution define convex sets, so they can be added to geometric programmes.

On status: Theorem 5.1 is a classical result, proved in Prékopa's papers (the chapter itself refers to Prékopa's survey for the proof). Prékopa's marginal theorem and the Prékopa–Leindler inequality are already machine-checked on this platform; Theorem 5.1 and the §5.2 and §5.7 theorems are, as far as a search of the platform shows, not formalized. The work is formalizing known proofs, in the log-concavity predicate the platform already uses.

Difficulty

The obvious argument for Theorem 5.1, "the constraint set is convex and the density is log-concave, so the probability is log-concave", hides the real content: log-concavity of a probability as a function of a parameter is a statement about integrals, and it does not follow from pointwise properties of the integrand without a Prékopa–Leindler-type inequality. Concavity of each $g_i$ separately in $x$ and in $y$ is not enough; joint concavity in $(x, y)$ is used essentially. The probability function vanishes on large regions in typical examples, so any argument that takes logarithms pointwise fails at the boundary of its support.

For Theorem 5.2.2 the naive argument fails at the index $i = 0$: nothing about $h_0$ is assumed directly, and log-concavity of $h_0$ is exactly Theorem 5.1. For Theorem 5.7.1 the difficulty is a sign condition on a covariance; without the ordering hypothesis the mixed derivative can be negative.

Formalization scope

$\mathbb R^n$ and $\mathbb R^q$ are EuclideanSpace ℝ (Fin n) and EuclideanSpace ℝ (Fin q); the constraint functions take pairs $(x, y)$ in the product, and concavity is ConcaveOn ℝ Set.univ on that product (joint concavity). A "continuous probability distribution with density $f$" is stated as P.map ξ = volume.withDensity (ENNReal.ofReal ∘ f) with $\xi$ and $f$ measurable and $P$ a probability measure. Log-concavity is the platform definition ConvexOptimization.LogConcaveOn (nonnegativity plus the power inequality), imported as a reference item; concavity of Real.log ∘ h₀ would be a different, wrong property because Lean's Real.log 0 = 0. Indices are 0-based (Fin r, Fin m, Fin N, Fin n); the probabilistic constraint $i = 0$ of Theorem 5.2.2 is stated separately from $h_1, \dots, h_m$. The polynomial distribution is a formula on Fin n → ℝ with real powers, used only on the cube.

No constant of the book is replaced by an explicit value: every result of this chapter is qualitative.

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

• Theorem 5.1 states the density condition "for every $x_1, x_2 \in \mathbb R^n$"; the density lives on $\mathbb R^q$ and the condition is imposed there.
• (5.19) prints the first factor as $z_i^{\alpha_{i1}}$ and the domain index as $i = 1, \dots, N$; they are read as $z_1^{\alpha_{i1}}$ and $j = 1, \dots, n$.
• Theorem 5.7.1 prints its ordering hypothesis with transposed indices ($\alpha_{11} \le \alpha_{12} \le \dots \le \alpha_{1n}$); following the proof, it is read as: across the $N$ terms the $z_1$-exponents increase and the $z_2$-exponents decrease.
• Theorem 5.7.1 claims "is a probability distribution function"; the book proves only (5.21), and normalization would need $\sum_i c_i = 1$, which is not assumed. The formal statement is (5.21), the mixed derivative as an iterated deriv.

Theorem 5.2.2 carries all six assumptions of §5.2, including compactness of the feasible set and the Slater point; it does not assume log-concavity of $h_0$, which must be derived from the density and concavity hypotheses. A trivializing formalization, such as one with a density hypothesis that no probability law satisfies or a log-concavity predicate that holds for every function vanishing somewhere, is ruled out: the density hypotheses are satisfiable (checked locally) and LogConcaveOn is the multiplicative inequality at every pair of points.

Needed infrastructure: Prékopa's marginal theorem (available), log-concavity of indicators of convex sets and of products, measurability of the constraint set, and Artin's theorem that a sum of log-convex functions is log-convex (for Theorem 5.7.2). Artin's theorem and closure properties of LogConcaveOn are reusable beyond this mission, and contributions of them are welcome.

Selected references

• A. Prékopa, "Numerical Solution of Probabilistic Constrained Programming Problems", in Yu. Ermoliev and R. J-B Wets (eds.), Numerical Techniques for Stochastic Optimization, Springer Series in Computational Mathematics 10, Springer 1988, Ch. 5, pp. 123–139. https://doi.org/10.1007/978-3-642-61370-8
• A. Prékopa, "Logarithmic concave measures with application to stochastic programming", Acta Sci. Math. (Szeged) 32 (1971), 301–316.
• A. Prékopa, "On logarithmic concave measures and functions", Acta Sci. Math. (Szeged) 34 (1973), 335–343.
• A. Charnes and W. W. Cooper, "Chance-constrained programming", Management Science 6 (1959), 73–79. https://doi.org/10.1287/mnsc.6.1.73
• B. L. Miller and H. M. Wagner, "Chance constrained programming with joint constraints", Operations Research 13 (1965), 930–945. https://doi.org/10.1287/opre.13.6.930
• A. Prékopa, Stochastic Programming, Kluwer 1995. https://doi.org/10.1007/978-94-017-3087-7

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),

$$Q(x,\xi)=\min\{q^Ty : Wy=h-Tx,\ y\ge 0\},\qquad \xi=(q,h,T),$$

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

$$E\varphi(\xi)\le\sum_{v\in\mathrm{vert}\,\Xi}\Big(\prod_{j=1}^m p_j(v_j)\Big)\varphi(v).$$

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

One-dimensional bin packing asks for the fewest unit-capacity bins that hold a given list of items with sizes in $(0,1)$. Deciding whether two bins suffice is NP-hard (it contains the partition problem), so no polynomial-time algorithm guarantees a ratio below $3/2$ unless P = NP. The natural question is therefore asymptotic: how small can the additive error $A(I) - OPT(I)$ be made, as a function of the optimum $OPT(I)$?

• 1974: D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey and R. L. Graham analysed First Fit and First Fit Decreasing, with asymptotic ratios $17/10$ and $11/9$ (SIAM J. Comput. 3(4)).
• 1981: W. Fernandez de la Vega and G. S. Lueker gave an asymptotic approximation scheme: for every $\varepsilon > 0$, $(1+\varepsilon)\,OPT(I) + 1$ bins in linear time (Combinatorica 1).
• 1982: N. Karmarkar and R. M. Karp replaced the multiplicative error by an additive one: $OPT(I) + O(\log^2 OPT(I))$ bins in polynomial time (Proc. 23rd FOCS). This mission formalizes that bound.
• 2017: R. Hoberg and T. Rothvoss improved the additive error to $O(\log OPT)$ (SODA 2017). Whether $OPT(I) + O(1)$ is achievable remains open.

Its main device, geometric grouping followed by rounding a linear program over bin configurations, recurs in later additive results and in cutting-stock problems.

Setting

An instance $I$ is a finite multiset of piece sizes in the open interval $(0,1)$. Write $n(I)$ for the number of pieces, $m(I)$ for the number of distinct sizes, $SIZE(I)$ for the total size and $a(I)$ for the smallest size. A packing is a multiset of bins whose union is $I$ and in each of which the sizes sum to at most $1$; its cost is the number of bins, and $OPT(I)$ is the least cost.

A configuration is a nonempty multiset of sizes occurring in $I$ that fits in one bin. The fractional bin-packing problem is the linear program

$$(I)\qquad \min\ \mathbf 1\cdot x\quad\text{s.t.}\quad x \ge 0,\quad Ax \ge b,$$

with one variable $x_j$ per configuration, where $A_{tj}$ counts the pieces of size $t$ in configuration $j$ and $b_t$ the pieces of size $t$ in $I$. Its value is $LIN(I)$. A basic feasible solution is an extreme point of the feasible region.

Geometric grouping with parameter $k$ sorts the pieces in non-increasing order and cuts them into consecutive groups $G_1, G_2, \dots, G_q$, each the shortest run of pieces of total size at least $k$. Within each group $G_i$ ($i \ge 2$) only as many of the largest pieces as $G_{i-1}$ has are kept; they are rounded up to the largest size in $G_i$, giving $G_i'$. The rounded pieces form $J$, and $G_1$ together with the unrounded leftovers $\Delta G_i$ form $J'$.

ALGORITHM 2 with a positive integer $k$ and a positive real $g$:

1. Eliminate all pieces of size $\le g$.
2. While $SIZE > 1 + \frac{1}{1-1/k}\ln\frac1g$: group the current instance into $J, J'$; pack $J'$ in at most $2k[2 + \ln\frac1g]$ bins; obtain a basic feasible solution $x$ of the LP of $J$ with cost at most $LIN(J)+1$; open $\lfloor x_j\rfloor$ bins of each configuration $j$, fill them with pieces, and delete the pieces so packed.
3. Pack the remaining pieces in at most $2 + \frac{2}{1-1/k}\ln\frac1g$ bins.
4. Reinsert the eliminated pieces, using a new bin only when necessary.

Its cost on $I$ is written $A(I)$.

Formalization targets

Goal: Theorem 4 with explicit constants

For every instance $I$ with $SIZE(I) \ge 2$, every packing that ALGORITHM 2 with $k=2$ and $g = 1/SIZE(I)$ can output is a packing of $I$ with

$$A(I) \le OPT(I) + \bigl(1 + \log_2 OPT(I)\bigr)\bigl(9 + 4\ln OPT(I)\bigr) + 2 + 4\ln OPT(I).$$

This is the paper's $A(I) \le OPT(I) + O(\log^2 OPT(I))$, with the constants that its proof yields.

The general bound for ALGORITHM 2

For integers $k \ge 2$, $0 < g \le \tfrac12$ and $SIZE(I) \ge 1$:

$$A(I) \le \max\Bigl\{(1+2g)\,OPT(I) + 1,\ OPT(I) + \Bigl[1 + \frac{\ln SIZE(I)}{\ln k}\Bigr]\Bigl[1 + 4k + 2k\ln\frac1g\Bigr] + 2 + \frac{2}{1-\frac1k}\ln\frac1g\Bigr\}.$$

Milestones

In attack order: Lemmas 1–3; Theorem 2 (items 1–3, the bound on $J'$, item 4 corrected); the per-iteration shrinking of $SIZE$; the bound on the number $t$ of iterations; the telescoping of $LIN$; the bin count after Step 3; the general bound.

Significance

The bound gives a polynomial-time algorithm whose additive error is polylogarithmic in the optimum, hence a fully polynomial asymptotic approximation scheme ($O(\log^2 OPT) = o(OPT)$). Varying $k$ and $g$ trades running time for error, as the paper notes after Theorem 4. The scheme of solving the rounded LP, keeping its integer part and re-grouping the residual is reused by later additive results, including the $O(\log OPT)$ bound of Hoberg and Rothvoss.

The result has been proved since 1982. No machine-checked proof of it, or of any bin-packing approximation guarantee of this kind, is known to exist in Lean or Mathlib. The mission produces a formal version whose hypotheses and constants are explicit. It also corrects two printed statements whose published forms are false: Theorem 2, item 4, and the chain of inequalities in the analysis that relies on it. The corrections are disclosed in the statements.

Difficulty

Rounding a single LP solution does not suffice. A basic solution of the configuration LP has at most $m$ fractional variables, and after rounding down, the leftover pieces form an instance of size at most $m(J)$. With linear grouping that leftover is of order $1/\varepsilon^2$ and costs a constant factor. The difficulty is making the residual shrink geometrically. Geometric grouping must produce an instance $J$ with $m(J) \le SIZE/k + O(\ln(1/g))$ distinct sizes while discarding only $O(k\ln(1/g))$ in $J'$. The residual must then be re-grouped and re-solved. Each step must be accounted for simultaneously in $SIZE$, $LIN$ and $OPT$, with an additive loss per iteration; the harmonic-sum estimate behind $SIZE(J')$ and the telescoping of $LIN$ across iterations carry most of the weight.

Formalization scope

• Model. An instance is a Multiset ℝ with sizes in the open interval $(0,1)$; real sizes generalize the paper's rationals, and the interval is open because a group of size at least $k$ must contain more than $k$ pieces. Packings are Multiset (Multiset ℝ). $OPT$ and $LIN$ are infima over nonempty sets. LP solutions are finitely supported functions on configurations; "basic" means extreme point.
• Subroutine contract. The Fractional Bin-Packing procedure is modelled only by its stated output: any basic feasible solution of cost at most $LIN(J)+1$. The ellipsoid method of §6 is not modelled.
• Runs. ALGORITHM 2 is a relation Alg2Run k g I P, witnessed by a trace. Every bound holds for every run: every admissible subroutine output, every packing at Steps 2 and 3 within the prescribed counts, every choice of pieces for the principal bins (which must fill every available slot), and every order of the Step 4 insertion. A separate well-definedness item states that a run exists, so the bounds are not vacuous.
• Explicit constants. $O(\log^2 OPT(I))$ in Theorem 4 is replaced by $(1+\log_2 OPT)(9 + 4\ln OPT) + 2 + 4\ln OPT$. The asymptotic threshold is made explicit as $SIZE(I) \ge 2$. $\ln$ is Real.log and $\log_2$ is Real.logb 2.
• Corrected statements. The last group of geometric grouping may fall short of $k$, which the paper ignores. For it, $\Delta G_q$ consists of the $\max(0, l_q - l_{q-1})$ smallest pieces. Theorem 2, item 4 is stated as $m(J) \le SIZE(J)/k + \ln(1/a(I)) + 1$; the printed version without $+1$ fails for $I = \{0.95, 0.95, 0.95, 0.9\}$, $k = 2$. Theorem 2 is stated for integers $k \ge 2$, which its proof needs. The iteration bound is stated for $t \ge 1$ and for the instance after Step 1.
• Out of scope. Running times, polynomiality, the function $T(m,n)$, the number of subroutine calls, §6, ALGORITHM 3 and Theorem 5.
• Ruling out trivial versions. "There exists a packing with at most $OPT(I) + \dots$ bins" is trivially true and is not the goal. The goal bounds every output of the algorithm, and the existence item shows that outputs exist.

Contributions welcome: milestone proofs; a harmonic-sum bound $\sum_{j=a}^{b} 1/j \le \ln\frac{b}{a-1}$; extreme-point facts for $\{x \ge 0 : Ax \ge b\}$ (at most as many nonzero coordinates as rows; an optimal extreme point exists), reusable beyond bin packing; monotonicity of $LIN$ and $OPT$ under the piecewise order.

Selected references

• N. Karmarkar, R. M. Karp, An Efficient Approximation Scheme for the One-Dimensional Bin-Packing Problem, Proc. 23rd Annual Symposium on Foundations of Computer Science (SFCS 1982), IEEE, pp. 312–320, 1982. https://doi.org/10.1109/SFCS.1982.61
• W. Fernandez de la Vega, G. S. Lueker, Bin packing can be solved within 1 + ε in linear time, Combinatorica 1(4), 349–355, 1981. https://doi.org/10.1007/BF02579456
• D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, R. L. Graham, Worst-case performance bounds for simple one-dimensional packing algorithms, SIAM J. Comput. 3(4), 299–325, 1974. https://doi.org/10.1137/0203025
• R. Hoberg, T. Rothvoss, A Logarithmic Additive Integrality Gap for Bin Packing, Proc. 28th ACM-SIAM SODA, 2616–2625, 2017. https://doi.org/10.1137/1.9781611974782.172