The OR Formalization Drive

Motivation

In a combinatorial auction a set of $m$ items is sold to $n$ bidders who value bundles of items rather than single items. Spectrum auctions, procurement of transportation lanes and the allocation of cloud resources all have this form, and the central algorithmic question is how to allocate the items so as to maximize the social welfare, the sum of the bidders' values for what they receive. Even to describe a general valuation takes $2^m$ numbers, so algorithms access the bidders through queries, and the achievable approximation depends on the class of valuations allowed.

Dobzinski, Nisan and Schapira (Math. Oper. Res. 35(1), 2010) study bidders without complementarities. For the class of XOS valuations, maxima of additive valuations, which strictly contains the submodular valuations, they give LP-based algorithms and, in §3.3, a purely combinatorial algorithm: bidders arrive one at a time, take their demanded bundle at the current item prices, and raise the prices of what they took. Its analysis charges the welfare of any allocation to the item prices the algorithm sets, an argument that uses only the demand and XOS oracles.

Setting

The items are $M=\{1,\dots,m\}$ and the bidders $1,\dots,n$.

• An additive valuation (a clause) $w$ assigns nonnegative values $w(1),\dots,w(m)$ to the items and $w(S)=\sum_{j\in S}w(j)$ to a bundle $S\subseteq M$.
• An XOS valuation $v$ is given by a nonempty finite set of clauses $\{w_1,\dots,w_t\}$ (its XOS expression) through $v(S)=\max_k w_k(S)$. A clause $w_k$ with $w_k(S)=v(S)$ is a maximizing clause for $S$. Every XOS valuation is normalized, $v(\emptyset)=0$, and monotone.
• An allocation is a tuple of pairwise disjoint bundles $O_1,\dots,O_n$; items may stay unallocated. Its welfare is $\sum_i v_i(O_i)$.
• A demand oracle for bidder $i$ answers, given item prices $p\in\mathbb R^m$, a bundle maximizing $v_i(T)-\sum_{j\in T}p_j$. An XOS oracle answers, given a bundle $S$, a maximizing clause for $S$ in $v_i$.

The greedy price-update algorithm. Start with all bundles empty and all prices $p_j=0$. For $i=1,\dots,n$: let $S_i$ be bidder $i$'s demand at the current prices; remove the items of $S_i$ from the bundles of the earlier bidders; let $q^i$ be the maximizing clause for $S_i$ in $v_i$; set $p_j=q^i_j$ for $j\in S_i$. Write $p^k$ for the prices after stage $k$ ($p^0=0$), $p^k(T)=\sum_{j\in T}p^k_j$, and $A_1,\dots,A_n$ for the final bundles.

In the Lean development these objects are XOSExpr, XOSExpr.val, IsAllocation, welfare, IsDemandOracle, IsXOSOracle, greedyState, greedyPrices and greedyAlloc in the namespace ComplementFreeCA.XOSGreedy.

Formalization targets

Goal: Theorem 3.3

For XOS valuations $v_1,\dots,v_n$, every demand oracle and every XOS oracle, and every allocation $O_1,\dots,O_n$,

$$\sum_{i=1}^n v_i(O_i)\;\le\;2\sum_{i=1}^n v_i(A_i).$$

The comparison with every allocation is the paper's comparison with the optimal allocation.

Milestones

1. Lemma 3.4. The final prices are paid for by the algorithm's welfare:

$$p^n(M)\le\sum_i v_i(A_i).$$

2. Lemma 3.5. Prices never decrease: $p^k_j\le p^{k'}_j$ for all items $j$ and stages $k\le k'$.
3. Lemma 3.6. For every allocation $O$,

$$\sum_i v_i(O_i)\le 2\,p^n(M).$$

Three further statements are included without being milestones: the output $A_1,\dots,A_n$ is an allocation (used implicitly by the paper); the first sentence of the proof of Lemma 3.6, that with $\Delta^i=p^i(M)-p^{i-1}(M)$ one has $\Delta^i=\max_{T\subseteq M}\bigl(v_i(T)-p^{i-1}(T)\bigr)$; and the factor $2$ is attained on the paper's two-item, two-bidder example (p. 9).

Significance

Theorem 3.3 shows that a factor-2 approximation of the optimal welfare for XOS bidders needs no linear program: one pass over the bidders, one demand query and one XOS query each. The paper's LP-based algorithm of §3.2 attains the better ratio $e/(e-1)$, and Theorem 4.1 of the paper shows that for the larger class of complement-free bidders no $(2-\epsilon)$-approximation is possible with polynomial communication.

The result is proved in the paper; to the best of our knowledge it has no machine-checked proof. This mission produces a formal model of XOS valuations, demand and XOS oracles and the greedy run that is reusable for other price-based arguments, and a formal proof of the guarantee for every tie-breaking in both oracles.

Difficulty

The argument is elementary, but its bookkeeping is where a formal proof can go wrong. Items move between bidders: an item taken from an earlier bidder in step (b) is re-priced in step (d), and every priced item lies in exactly one final bundle. Lemma 3.4 needs this invariant across all $n$ stages, together with the fact that a maximizing clause for the demanded set bounds $v_i$ on every subset, which holds because it is a clause of $v_i$ itself, not merely an additive function agreeing with $v_i$ on $S_i$. Lemma 3.5 is a contradiction argument using optimality of the demand; Lemma 3.6 telescopes the price increases and needs prices to stay nonnegative. The argument must hold for arbitrary oracle answers, so no canonical demand can be assumed.

Formalization scope

• Bidders are Fin n, items Fin m, bundles Finset (Fin m), values in ℝ. An XOS valuation is represented by its expression: a nonempty Finset (Fin m → ℝ) of clauses with nonnegative entries, evaluated with Finset.sup'. The paper's standing assumptions, normalized and monotone valuations (p. 1), hold automatically in this representation.
• The oracles are function parameters dem : Fin n → (Fin m → ℝ) → Finset (Fin m) and cl : Fin n → Finset (Fin m) → (Fin m → ℝ) with hypotheses IsDemandOracle and IsXOSOracle; every theorem holds for every such pair, so no tie-breaking rule is fixed.
• The run is a recursion on the stage: stage $k+1$ processes the 0-based bidder $k$, i.e. the paper's bidder $k+1$; after stage $n$ the state is constant.
• The paper's goal compares with the optimal allocation; the Lean goal compares with every allocation, which is equivalent and avoids an argmax. Stating the bound against one particular allocation, or against the algorithm's own output, would be trivial and is excluded.
• There are no $O(\cdot)$ constants: the factor $2$ is the paper's.
• Printed slip: the model on p. 1 writes an allocation as $S_1,\dots,S_m$; it is one bundle per bidder, $S_1,\dots,S_n$.
• Out of scope: running time, the cost of simulating oracles (Proposition 2.1), and communication lower bounds.

Contributions welcome: proofs of the three milestones and of the goal, and reusable lemmas on the invariant that every priced item lies in exactly one final bundle.

Selected references

• S. Dobzinski, N. Nisan, M. Schapira, Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders, Mathematics of Operations Research 35(1):1–13, 2010. https://doi.org/10.1287/moor.1090.0436
• B. Lehmann, D. Lehmann, N. Nisan, Combinatorial auctions with decreasing marginal utilities, Games and Economic Behavior 55(2):270–296, 2006. https://doi.org/10.1016/j.geb.2005.02.006

Motivation

In a combinatorial auction a seller offers several indivisible items at once, and bidders value bundles of items rather than items one at a time. Allocating the items to maximize total value is the central optimization problem of the area, and it arises in spectrum licensing, procurement and transport contracting (Cramton, Shoham and Steinberg, Combinatorial Auctions, MIT Press, 2006). Two obstacles meet. Computationally, a valuation has $2^m$ numbers, so an algorithm can only query it, and even then optimization is hard. Strategically, the valuations are private: a bidder reports whatever maximizes its own utility, so an algorithm that is a good approximation on true inputs may be useless on reported ones.

The classical answer to the strategic obstacle is the VCG payment scheme, which makes truthful reporting a dominant strategy but requires the exact optimum. Nisan and Ronen (2007) showed that an approximation algorithm becomes truthful under VCG payments essentially only when it is maximal in range: it fixes a restricted set of allocations in advance and optimizes exactly over that set. Dobzinski, Nisan and Schapira (Math. Oper. Res. 35(1), 2010, §5) give such an algorithm for complement-free (subadditive) bidders that uses only value queries and loses a factor of order $\sqrt m$. For general valuations in the value-query model the paper cites a lower bound of order $m/\log m$ (Dobzinski and Schapira, working paper 2005; Blumrosen and Nisan, Hebrew University Discussion Paper 381, 2005; see the paper's references [7] and [2]), and the same paper (Theorem 6.1) shows that even XOS bidders cannot be approximated within $m^{1/2-\epsilon}$ with polynomially many value queries.

Setting

A set $M=\{1,\dots,m\}$ of items is sold to $n$ bidders. Bidder $i$ has a valuation $v_i$ that assigns a real number $v_i(S)$ to every bundle $S\subseteq M$. Throughout, valuations are normalized, $v_i(\emptyset)=0$, and monotone, $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$. A valuation is complement free (CF) if $v(S\cup T)\le v(S)+v(T)$ for all bundles $S,T$. An allocation $A=(A_1,\dots,A_n)$ gives the bidders pairwise disjoint bundles (items may stay unallocated), and its social welfare is $\sum_i v_i(A_i)$.

The mechanism receives reports $b=(b_1,\dots,b_n)$ and runs the following algorithm $\mathrm{ALG}$:

1. query $b_i(M)$ and $b_i(\{j\})$ for every bidder $i$ and item $j$;
2. compute a maximum-weight matching $P$ in the complete bipartite graph between items and bidders, where the edge between item $j$ and bidder $i$ costs $b_i(\{j\})$;
3. if the bidder $t$ maximizing $b_i(M)$ has $b_t(M)$ strictly larger than the weight $|P|$, give all items to $t$; otherwise give every item matched by $P$ to its matched bidder.

Its range $R$ is the set of allocations that give all of $M$ to one bidder, together with the allocations in which every bidder receives at most one item. Under VCG payments bidder $i$ receives $\sum_{k\ne i}b_k(\mathrm{ALG}(b)_k)$, so its utility is $v_i(\mathrm{ALG}(b)_i)+\sum_{k\ne i}b_k(\mathrm{ALG}(b)_k)$. The mechanism is incentive compatible on a class of valuations if no bidder can raise its utility by misreporting within that class, whatever the others report.

Formalization targets

Goal: Theorem 5.1 (p. 11)

For every choice of the maximum-weight matching and of the top bidder as functions of the reports, for every profile $v$ of normalized, monotone, CF valuations and every allocation $O$,

$$\sum_{i=1}^n v_i(O_i)\;\le\;2\sqrt m\,\sum_{i=1}^n v_i\big(\mathrm{ALG}(v)_i\big),$$

and the mechanism $(\mathrm{ALG},\text{VCG payments})$ is incentive compatible on the CF valuations.

Milestones, in attack order

1. §5.1, VCG. Welfare maximization with Groves payments is incentive compatible (a published platform theorem, AGT.vcg_incentive_compatible).
2. §5.1, maximal in range. Any allocation rule that optimizes reported welfare exactly over a fixed range is incentive compatible under VCG payments on the same domain.
3. ALG is maximal in range with range $R$ on normalized reports.
4. The CF single-item bound. For a CF valuation and $c\in T$ maximizing $v(\{j\})$ over $T$: $v(T)\le\sum_{j\in T}v(\{j\})\le|T|\,v(\{c\})$.
5. First case. If bidders with $|O_i|\ge\sqrt m$ carry at least half the welfare of $O$, then $\sum_i v_i(O_i)\le 2\sqrt m\,v_t(M)$ for the top bidder $t$.
6. Second case. Otherwise some allocation in which every bidder gets at most one item has welfare at least $\sum_i v_i(O_i)/(2\sqrt m)$.

Significance

The theorem shows that, for subadditive bidders, the $\sqrt m$ barrier known for general valuations can be matched by a truthful mechanism that asks each bidder only $m+1$ value queries. It is one of the early examples of maximal-in-range mechanism design, a template later used for many truthful approximation mechanisms in combinatorial auctions, and it sits against Theorem 6.1 of the same paper, which shows that for XOS bidders no value-query algorithm with polynomially many queries does better than $m^{1/2-\epsilon}$.

The result is proved in the paper. What this mission adds is a machine-checked proof: a formal model of VCG-based mechanisms over a restricted range, a proof that the §5.2 algorithm is maximal in range for every tie-breaking of its two optimization steps, and the explicit constant $2$ in the $O(\sqrt m)$ bound. To our knowledge neither half of Theorem 5.1 is formalized elsewhere; the general VCG theorem exists on the platform in the setting of arbitrary outcome sets.

Difficulty

The approximation argument partitions the bidders of a reference allocation by whether their bundles have at least $\sqrt m$ items, and the two cases need different facts: disjointness bounds the number of large bundles by $\sqrt m$, and subadditivity bounds each small bundle by its size times its best item. A naive transcription breaks at degenerate inputs: the page divides by $|T_i|$ and writes strict inequalities, both of which fail when a bundle is empty or all values are zero, so the formal statement must be organized around non-strict bounds.

Incentive compatibility has a different obstacle. It holds only if the allocation rule depends on the reports alone and optimizes exactly over its range, including at ties between the grand bundle and the matching. The matching and the top bidder are not unique, so the proof must work for an arbitrary but fixed tie-breaking, and the welfare of the matching allocation must be identified with the matching weight, which uses normalization of every bidder who receives nothing.

Formalization scope

Bidders are Fin n, items Fin m, bundles Finset (Fin m), valuations Finset (Fin m) → ℝ. Normalization and monotonicity (the paper's standing assumptions, p. 1) and complement freedom are hypotheses; IsCFValuation bundles all three. An allocation is a family of pairwise disjoint bundles; unallocated items are allowed. A matching is a partial map Fin m → Option (Fin n) with no bidder matched twice.

Conventions the formalization commits to:

• Explicit constant. The paper writes $O(\sqrt m)$; its proof yields $2\sqrt m$ (both cases end with $|OPT|/(2\sqrt m)$), and the goal states $2\sqrt m$ with Real.sqrt m.
• Oracles and ties. The maximum-weight matching and the top bidder enter as functions mat, top of the report profile, each with a specification hypothesis; the goal is stated for every such pair. The algorithm reads only the reports; the tie between $b_t(M)$ and $|P|$ goes to the matching, as on the page.
• Payments. The mechanism pays each bidder $\sum_{k\ne i}b_k(\cdot)$, the paper's convention (footnote 2, p. 11); incentive compatibility is stated on the CF domain, the paper's. The local definition mirrors AGT.MechIncentiveCompatible on outcomes $a\mapsto v_i(a_i)$.
• Reference allocation. The approximation is stated against every allocation $O$, not only an optimal one; this is equivalent and avoids a junk maximum.
• Printed slips. The strict inequalities and the division by $|T_i|$ in the second case are replaced by non-strict, multiplied forms; the first case concludes for a bidder maximizing $v_i(M)$ rather than $v_i(O_i)$.
• Degenerate sizes. At $m=0$ everything is zero and the bound holds trivially; with $n=0$ no top-bidder rule exists.
• Out of scope. "In polynomial time" is a running-time claim and is not modelled.

A trivializing formalization is ruled out: the ratio is the explicit $2\sqrt m$ rather than an existential constant, incentive compatibility is over the full CF domain (not additive reports only) for a rule that cannot see true valuations, and the rules mat, top are satisfiable (a maximum over the finitely many matchings exists; a top bidder exists when $n\ge1$).

Useful infrastructure: finite maximum-weight matchings on complete bipartite graphs, subadditivity bounds over Finset sums, and a reusable lemma that maximal-in-range rules with VCG payments are truthful. Contributions of any milestone are welcome; milestones 2 and 4 are self-contained.

Selected references

• S. Dobzinski, N. Nisan, M. Schapira, Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders, Mathematics of Operations Research 35(1):1–13, 2010. https://doi.org/10.1287/moor.1090.0436
• N. Nisan, A. Ronen, Computationally Feasible VCG Mechanisms, Journal of Artificial Intelligence Research 29:19–47, 2007. https://doi.org/10.1613/jair.2046
• S. Dobzinski, M. Schapira, Optimal Upper and Lower Approximation Bounds for k-Duplicates Combinatorial Auctions, working paper, The Hebrew University of Jerusalem, 2005 (reference [7] of the paper).
• L. Blumrosen, N. Nisan, On the Computational Power of Iterative Auctions I: Demand Queries, Discussion Paper 381, Center for the Study of Rationality, The Hebrew University of Jerusalem, 2005 (reference [2] of the paper).
• N. Nisan, Introduction to Mechanism Design (for Computer Scientists), in N. Nisan, T. Roughgarden, E. Tardos, V. Vazirani (eds.), Algorithmic Game Theory, Cambridge University Press, 2007, pp. 209–242.
• P. Cramton, Y. Shoham, R. Steinberg (eds.), Combinatorial Auctions, MIT Press, 2006.

Motivation

Insertion heuristics build a traveling salesman tour one city at a time: start from a single city, and at each step choose a city not yet on the subtour and splice it into the subtour where it lengthens the subtour least. Local search heuristics start from a tour and repeatedly replace a few of its edges by others while this shortens the tour. Both families are standard in practice, and a natural engineering idea is to combine them: run an insertion heuristic, then polish the result by local search. The question this mission formalizes is whether local optimality of the insertion tour certifies anything about its quality.

Rosenkrantz, Stearns and Lewis (SIAM J. Comput. 6(3), 1977) answered this for graphs satisfying the triangle inequality. Their §4 proves that nearest and cheapest insertion always return a tour of length at most $2(1-1/n)$ times the optimal length, and their Theorem 5 shows this bound is attained. Their §7 then shows that the very tour attaining the bound is $k$-optimal for every $k\le n/4$: no exchange of $k$ edges shortens it. So the insertion bound is tight even for tours that local search with $k$-changes cannot improve.

Timeline, as far as this mission is concerned:

• 1965: Lin (Bell System Tech. J. 44) defines $k$-optimal tours and uses 3-optimal local search.
• 1973: Lin and Kernighan (Oper. Res. 21) generalize the edge-exchange neighbourhoods.
• 1977: Rosenkrantz, Stearns and Lewis prove the $2(1-1/n)$ upper bound for nearest and cheapest insertion (Theorem 4 and its corollary), its tightness for $n\ge 6$ (Theorem 5), the existence of $k$-optimal tours with the same ratio (Theorem 6, stated for $n\ge 8$), and the Corollary combining the two.

Setting

A traveling salesman graph on $n$ nodes is the node set $N=\{1,\dots,n\}$ with a distance $d(i,j)\ge 0$ that is symmetric and satisfies the triangle inequality $d(i,k)\le d(i,j)+d(j,k)$. A tour is a Hamiltonian circuit; its length is the sum of its edge lengths; OPTIMAL is the least length of a tour. As in the paper, the identically zero distance is excluded, so OPTIMAL $>0$.

A subtour is a circuit on a subset of the nodes (a single node is a subtour without edges). For a subtour $T$ and a node $k\notin T$, TOUR$(T,k)$ is obtained by deleting an edge $(x,y)$ of $T$ minimizing $d(x,k)+d(k,y)-d(x,y)$ and adding $(x,k)$ and $(k,y)$; COST$(T,k)$ is the resulting increase in length. An insertion method chooses nodes $a_0,a_1,\dots,a_{n-1}$, starts from $T_1=\{a_0\}$ and sets $T_{i+1}=\mathrm{TOUR}(T_i,a_i)$; INSERT is the length of $T_n$. Nearest insertion chooses $a_i$ minimizing $d(T_i,x)=\min_{y\in T_i}d(y,x)$ over $x\notin T_i$; cheapest insertion chooses $a_i$ minimizing COST$(T_i,x)$. Ties are broken arbitrarily.

A $k$-change of a tour deletes $k$ of its edges and adds $k$ other edges so that another tour is obtained. A tour is $k$-optimal if no $k$-change produces a strictly shorter tour.

The extremal instance is the circle $(N_n,d_n)$: $n$ cities equally spaced on a circular road, with $d_n(i,j)$ the smallest $m\ge 0$ with $i-j\equiv m$ or $j-i\equiv m \pmod n$. The insertion run of Theorem 5 inserts the cities in the order $1,2,\dots,n$ and produces the zig-zag tour $T_n$: city 1, then the even cities in increasing order, then the odd cities in decreasing order.

Formalization targets

Goal: the Corollary to Theorem 6

For $n\ge 6$ and $4k\le n$ there is a traveling salesman graph with OPTIMAL $>0$ on which some run of nearest insertion, and some run of cheapest insertion, return a $k$-optimal tour with

$$\frac{\mathrm{INSERT}}{\mathrm{OPTIMAL}}=2\left(1-\frac1n\right).$$

Milestones

1. The insertion run on the circle: the subtours $T_i$ and nodes $a_i=i+1$ form an insertion run that obeys both the nearest and the cheapest rule (proof of Theorem 5).
2. On the circle, $T_n$ has length $2(n-1)$ and OPTIMAL $=n$ (proof of Theorem 5).
3. Theorem 5: for $n\ge 6$ there is a graph with INSERT/OPTIMAL $=2(1-1/n)$ for both methods.
4. Equation (7.4): the length of a tour of the circle is the sum over unit edges $e$ of COUNT$(e,T)$, the number of times $e$ is traversed when each tour edge is replaced by a shortest arc.
5. Every tour of the circle is odd or even (all counts of one parity), eq. (7.5).
6. $T_n$ is the shortest even tour, so every tour shorter than $T_n$ is odd.
7. $T_n$ is $k$-optimal for every $k\le n/4$.
8. Theorem 6: for $n\ge 8$ there is a graph with a tour that is $k$-optimal for all $k\le n/4$ and has LOCALOPT/OPTIMAL $=2(1-1/n)$.

Significance

The result. The Corollary shows that the $2(1-1/n)$ worst-case guarantee of nearest and cheapest insertion cannot be improved by requiring that the returned tour survive $k$-change local search, for $k$ up to a quarter of the number of cities. Theorem 6 says more generally that $k$-optimality with $k\le n/4$ does not bound the ratio to the optimum below $2(1-1/n)$. Together with the paper's upper bound, the insertion guarantee is exact, and it stays exact after local polishing with small neighbourhoods.

Formalizing it. All statements are proved in the paper; none, to our knowledge, has been machine-checked. The platform has an upper bound for nearest insertion (SupplyChainTheory, Theorem 10.7, ratio at most 2) but no tightness example and no notion of $k$-optimality. This mission produces a reusable definition of $k$-changes against arbitrary tours, the circle metric, and the parity-counting argument on a cycle, and it supplies the calculations the paper omits ("We omit these calculations but note that they require the assumption $n\ge 6$").

Difficulty

Two steps carry the weight. First, the omitted calculations for the insertion run: at each stage one must show that inserting $a_i$ between $i-1$ and $i$ minimizes the insertion increase over every edge of the zig-zag subtour, and that no other outside city can be inserted for less than 2; the claim fails for $n=4$ and $n=5$, so the verification must use $n\ge 6$ in an essential way. Second, $k$-optimality is a statement about every tour at edge difference $k$, not about 2-opt segment reversals or any specific move family. A search over moves of a special form does not establish it; the argument must bound the length of an arbitrary tour at edge difference $k$ from below.

Formalization scope

Nodes are Fin n, so the paper's node $m$ is index $m-1$ and $a_i=i+1$ is index $i$; subtour indices stay 1-based ($T_1=[a_0]$, approximation $T_n$). A distance is d : Fin n → Fin n → ℝ with the structure IsTSPDist (symmetric, nonnegative, triangle inequality, and $d(i,i)=0$; the last is a normalization absent from the paper that changes no length). A tour is an Equiv.Perm (Fin n), a subtour a list read cyclically; OPTIMAL is Finset.univ.inf' over permutations, the true minimum. TOUR$(T,k)$ is insertion at a position minimizing the new length; COST is a minimum over positions; the nearest-insertion distance (4.1) takes values in WithTop ℝ, so no junk value arises. $k$-optimality compares the tour with every permutation whose edge set (unordered pairs) misses exactly $k$ of the tour's edges. Ratios are multiplied out.

COUNT$(e,T)$ needs a choice of shortest arc for antipodal pairs when $n$ is even; the formalization takes the arc through $\min(x,y),\dots,\max(x,y)$. The paper's argument does not depend on this choice.

Deviations from the printed text: the Corollary is stated for $n\ge 6$ (the printed statement says only $4k\le n$, but its proof uses the example of Theorem 5, which exists for $n\ge 6$; for $k=0$, $n=3$ the printed statement is false). Theorem 6 keeps its printed $n\ge 8$. In the proof of Theorem 5 the paper writes "(4.2) holds" where the cheapest-insertion condition (4.3) is meant; the formal statement uses (4.3).

The existence statements carry OPTIMAL $>0$, the paper's standing assumption (1.1). Without it, the zero distance would make every length zero and every tour $k$-optimal, which would satisfy the ratio equations trivially; that formalization is ruled out.

Contributions welcome: proofs of any milestone, in particular the omitted insertion calculations and the parity lemma, and reusable lemmas on cyclic lists and edge sets of permutations.

Selected references

• D. J. Rosenkrantz, R. E. Stearns, P. M. Lewis II, An Analysis of Several Heuristics for the Traveling Salesman Problem, SIAM J. Comput. 6(3):563–581, 1977. https://doi.org/10.1137/0206041
• S. Lin, Computer solutions of the traveling salesman problem, Bell System Tech. J. 44:2245–2269, 1965. https://doi.org/10.1002/j.1538-7305.1965.tb04146.x
• S. Lin, B. W. Kernighan, An effective heuristic algorithm for the traveling-salesman problem, Oper. Res. 21(2):498–516, 1973. https://doi.org/10.1287/opre.21.2.498

Motivation

This mission completes chapter 7's L-convex function theory and closes the book on it. It first finishes the theory of positively homogeneous L-convex functions — showing they coincide exactly with the classical Lovász extensions of submodular set functions, a one-to-one correspondence that recognizes forty years of submodular-optimization machinery as a special case of L-convex function theory. It then proves the L-side capstone this series has been building toward since mission 26-ch07b-lconvexfunctions: polyhedral L-convexity is characterized simultaneously by directional derivatives, subdifferentials, and weighted-minimizer polyhedra — the exact mirror of what mission 24-ch06d-mconvexfunctions proved for M-convex functions. Finally it develops quasi L-convex functions, the L-side analogue of mission 25-ch06e-mconvexfunctions's quasi M-convex functions, ending exactly where chapter 7 itself ends.

Setting

Fix a finite ground set $V$. The class $0L[\mathbb R \to \mathbb R]$ consists of polyhedral L-convex functions that are positively homogeneous; $0L[\mathbb Z \to \mathbb Z]$, its integer-valued integer-domain analogue. A positively homogeneous L-convex function $g$ induces a submodular set function $\rho_g(X) = g(\chi_X)$; conversely the Lovász extension $\hat\rho$ of a submodular set function is positively homogeneous L-convex. The base polyhedron $B(\rho) = \{x \in \mathbb R^V : x(X) \le \rho(X)\ \forall X,\ x(V) = \rho(V)\}$ of a submodular $\rho$ is always an M-convex polyhedron. A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is quasi submodular (QSB) if $g(p\wedge q) \le g(p)$ or $g(p\vee q) \le g(q)$ for all $p,q$; the weaker (QSBw) requires only $\max\{g(p),g(q)\} \ge \min\{g(p\wedge q), g(p\vee q)\}$.

Formalization targets

Goal: the four characterizations of polyhedral L-convexity (Theorem 7.45)

For a polyhedral convex function $g$ with $\operatorname{dom}_{\mathbb R} g \ne \emptyset$: $g$ is L-convex if and only if every directional derivative $g'(p;\cdot)$ is $0L[\mathbb R \to \mathbb R]$, if and only if every subdifferential $\partial_{\mathbb R} g(p)$ is an M-convex polyhedron, if and only if every weighted minimizer set is an L-convex polyhedron. Not present in this chunk's own extraction table (its label opens mid-paragraph, missed by the same extractor failure already documented for mission 25-ch06e-mconvexfunctions's Theorem 6.68), found by direct reading and chosen as goal because it is the exact L-side mirror of mission 24-ch06d-mconvexfunctions's own milestone Theorem 6.63, and its proof is assembled entirely from results already in this series (Theorem 7.43, Proposition 7.34, Theorem 7.40).

Supporting structural targets

Fifteen further results build the two remaining pieces of chapter 7's theory. Propositions 7.37-7.39 and Theorem 7.40 establish the one-to-one correspondence between positively homogeneous L-convex functions and submodular set functions via the Lovász extension; Proposition 7.41 gives a minimizer-polyhedron characterization of this class, and Proposition 7.42 shows directional derivatives of L-convex functions automatically land in it. Theorem 7.43 — the L-side mirror of mission 24-ch06d-mconvexfunctions's own goal, Theorem 6.61 — proves the directional- derivative/subdifferential correspondence via the induced submodular set function's base polyhedron; Proposition 7.44 checks consistency at integer points, and Theorem 7.46 refines Theorem 7.45 to the integral case. Theorem 7.49 (also missed by the extractor) gives the quasi-submodularity implication hierarchy and its perturbation-equivalence capstone, mirroring mission 25-ch06e-mconvexfunctions's Theorem 6.68 exactly. Proposition 7.50 and Theorems 7.51-7.52 build the level-set/perturbation machinery quasi submodularity needs; Theorems 7.53-7.54 (both missed by the extractor, the latter's full statement requiring one page beyond this chunk's nominal range, at the very end of chapter 7) give the quasi L-optimality and quasi L-proximity theorems.

Significance

The 0L/submodular correspondence (Theorem 7.40) is the precise sense in which L-convex function theory generalizes submodular set function theory rather than merely resembling it: every submodular set function is literally the restriction to $\{0,1\}^V$ of a positively homogeneous L-convex function, and every algorithm for one transfers to the other through this exact dictionary. The goal, Theorem 7.45, completes the parallel structure this series has built since chapter 6: M-convexity and L-convexity are now each characterized in the same four convex-analytic vocabularies, setting up chapter 8's conjugacy theorem, which will show these two characterizations are not merely analogous but literally dual to each other under the Legendre-Fenchel transform. The quasi-submodularity results matter for the same reason as their M-side counterparts: chapter 10's algorithms for L-convex-function minimization remain correct under nonlinear rescalings that destroy L-convexity itself but preserve quasi submodularity.

None of these results are open — they are Murota's account of how far L-convex function theory extends beyond the polyhedral case (to positive homogeneity and its submodular-function incarnation) and how far its exchange-style inequality can be relaxed while preserving optimization theory (to quasi submodularity), mirroring chapter 6's identical two-part program for M-convex functions. What this mission contributes is a faithful, machine-checked formal statement of each, including four theorems (7.45, 7.49, 7.53, 7.54) the platform's own automated extractor missed entirely — one of them requiring a page beyond this chunk's own nominal range to complete, since chapter 7 ends there and this is the last mission covering it — extending the shared Lean vocabulary (ZeroLR, BasePolyhedron, QSBw) this series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to prove all six pairwise implications among its four conditions independently; the book's own proof instead chains through results already established: (a)⇒(b) is Proposition 7.42, (a)⇒(c) is Theorem 7.43, (a)⇒(d) is Proposition 7.34, (b)⇔(c) uses the 0L/M0[R] correspondence, and (d)⇒(b) is the genuinely hard direction, requiring Proposition 7.41 applied to the directional derivative itself (showing $\arg\min(g'(p;\cdot)[-x])$ is an L-convex cone by an explicit description via the admissible-potential set of the distance function underlying $\arg\min g[-x]$). The remaining combinatorial difficulty in this block is in Theorem 7.43's proof: identifying $\partial_{\mathbb R} g(p)$ with the base polyhedron $B(\rho_{g,p})$ requires the L-optimality criterion (Theorem 7.33, mission 27-ch07c-lconvexfunctions) applied pointwise, a chain of logical equivalences with no single-step shortcut, exactly mirroring how mission 24-ch06d-mconvexfunctions's Theorem 6.61 needed the M-optimality criterion.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; L-convex functions are (V→ℝ)→WithTop ℝ (polyhedral) or (V→ℤ)→WithTop ℝ (integer-domain). All sixteen numbered results found in this chunk's page range — the twelve in BRIEF.md's own table plus four the extractor missed (Theorems 7.45, 7.49, 7.53, 7.54) — are placed, with two documented, content-preserving scope decisions: Theorem 7.43 omits the dual-integral refinement clauses for L[R→R|Z]/L[Z→Z] (the same decision mission 24-ch06d-mconvexfunctions's Theorem 6.61 made), and Proposition 7.50 states the general inequalities without restating their "In particular" specializations, which add no independent content — see HARD.md. "$\inf g[-x]>-\infty$" is replaced by the equivalent (ArgMinR ...).Nonempty hypothesis throughout, matching mission 25-ch06e-mconvexfunctions's identical substitution. This mission's base vocabulary is redeclared from missions 20-ch04b-mconvexsets, 21-ch05b-lconvexsets, 23-24-ch06*-mconvexfunctions, and 26-27-ch07*-lconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the sixteen sorrys are welcome; the goal and Theorem 7.43 carry the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• K. Murota and A. Shioura, "M-convex function on generalized polymatroid," Mathematics of Operations Research, 24 (1999), pp. 95-105 [152] (the polyhedral theory Theorems 7.26-7.46 are drawn from).
• P. Milgrom and C. Shannon, "Monotone comparative statics," Econometrica, 62 (1994), pp. 157-180 [129] (the origin of the quasi-submodularity condition (SSQSB)).

Motivation

A retailer or an airline decides which products to make available, and customers choose among what is offered. When a preferred product is missing, many customers substitute to another product instead of leaving. Assortment optimization asks which subset of products to offer so that the expected revenue from a customer is as large as possible, and its answer depends entirely on the choice model used to describe substitution.

The Markov chain choice model was introduced by Blanchet, Gallego and Goyal (EC 2013; Oper. Res. 64(4), 2016), who showed that it contains the multinomial logit model as a special case and proposed it as an approximation of general random-utility choice models. Feldman and Topaloglu (Oper. Res. 65(5), 2017) study revenue management under this model. Their first result, the subject of this mission, is that the assortment problem, a search over all $2^n$ offer sets, is solved by one linear program with $n$ variables. The same paper uses this result for its dynamic single-resource and network results, which are the subjects of the companion missions II–IV of this series.

Timeline:

• 2013/2016: Blanchet, Gallego and Goyal introduce the model and give a polynomial-time assortment algorithm.
• 2017: Feldman and Topaloglu show that the optimal assortment is read off an optimal solution of a dual linear program (their Theorem 2), and derive structural and capacity-control consequences.

Setting

There are $n$ products $N=\{1,\dots,n\}$. A customer arrives to purchase product $j$ with probability $\lambda_j$. If the product she visits is offered, she buys it. Otherwise she transitions to product $i$ with probability $\rho_{j,i}$ and checks whether $i$ is offered, or leaves without buying with probability $1-\sum_{i\in N}\rho_{j,i}$. Throughout, $\lambda_j>0$, $\rho_{j,i}\ge0$ and $\sum_{i\in N}\rho_{j,i}<1$ for all $j$.

For an offer set $S\subseteq N$, let $P_{j,S}$ be the expected number of visits to product $j$ while it is offered (the probability that $j$ is purchased) and $R_{j,S}$ the expected number of visits to $j$ while it is not offered. The pair $(P_S,R_S)$ is the unique solution of the (Balance) equations

$$P_{j,S}+R_{j,S}=\lambda_j+\sum_{i\in N}\rho_{i,j}R_{i,S}\ \ \forall j\in N,\qquad P_{j,S}=0\ \ \forall j\notin S,\qquad R_{j,S}=0\ \ \forall j\in S .$$

With revenue $r_j\in\mathbb R$ for product $j$, the (Assortment) problem is $\max_{S\subseteq N}\sum_{j\in N}P_{j,S}r_j$. Two linear programs enter: the maximization of $\sum_j r_jx_j$ over the polyhedron

$$\mathcal H=\Big\{(x,z)\in\mathbb R^{2n}_+:\ x_j+z_j=\lambda_j+\sum_{i\in N}\rho_{i,j}z_i\ \ \forall j\in N\Big\},$$

and its dual

$$\min_{v\in\mathbb R^n}\Big\{\sum_{j\in N}\lambda_jv_j:\ v_j\ge r_j\ \ \forall j\in N,\ \ v_j\ge\sum_{i\in N}\rho_{j,i}v_i\ \ \forall j\in N\Big\}.\qquad\text{(Dual)}$$

Formalization targets

Goal: Theorem 2 (p. 1326)

For every optimal solution $\hat v$ of (Dual), the set $\hat S=\{j\in N:\hat v_j=r_j\}$ is an optimal assortment:

$$\sum_{j\in N}P_{j,S}\,r_j\ \le\ \sum_{j\in N}P_{j,\hat S}\,r_j\qquad\text{for all } S\subseteq N .$$

Milestones, in the order the paper uses them

1. (p. 1325) The (Balance) equations have a unique nonnegative solution for every $S$.
2. Lemma 1 (p. 1326): for an extreme point $(\hat x,\hat z)$ of $\mathcal H$ and $S_{\hat x}=\{j:\hat x_j>0\}$, $P_{j,S_{\hat x}}=\hat x_j$ and $R_{j,S_{\hat x}}=\hat z_j$ for all $j$.
3. (p. 1326) The linear program over $\mathcal H$ has an optimal solution, and its optimal value equals the optimal value of (Assortment).
4. (p. 1326) (Dual) has an optimal solution, and its optimal value equals that of the linear program over $\mathcal H$.
5. (p. 1326, proof of Theorem 2) An optimal $\hat v$ satisfies $\hat v_j=r_j$ or $\hat v_j=\sum_i\rho_{j,i}\hat v_i$ for each $j$.

Significance

Theorem 2 reduces a combinatorial problem over $2^n$ offer sets to a linear program, so the assortment problem under the Markov chain choice model is solvable in polynomial time. The same structure drives the rest of the paper: the dual variables $\hat v_j$ are used to show that optimal offer sets shrink when all revenues fall by a common amount, to show that the optimal offer sets of the single-resource dynamic program are nested in the remaining capacity and time, and to reduce the choice-based network linear program to a compact one.

The result is proved in the paper. This mission produces a machine-checked version of it and of its supporting lemmas; to the knowledge of the mission author no formalization of the Markov chain choice model exists. The definition layer (the model, the (Balance) solution, $\mathcal H$ and (Dual)) is the first formal encoding of this choice model and is shared, with the same encoding, by missions II–IV.

Difficulty

The obvious route, comparing $\sum_jP_{j,S}r_j$ across offer sets directly, fails because $P_{j,S}$ is defined only implicitly through a linear system whose coefficient matrix changes with $S$; there is no closed-form expression that can be compared across offer sets, and enumerating the $2^n$ sets is exponential. The connection to linear programming needs an exact correspondence between the vertices of $\mathcal H$ and the (Balance) solutions, which is a statement about polyhedra, not about Markov chains. Even well-posedness is not free: existence, uniqueness and nonnegativity of $(P_S,R_S)$ depend on the row sums $\sum_i\rho_{j,i}$ being strictly below one. Mathlib has extreme points of convex sets but no ready-made theory of vertices of polyhedra or of linear programming duality in this form.

Formalization scope

Products are Fin n (0-based), offer sets are Finset (Fin n) (the paper's $S\subset N$ is non-strict inclusion, so every subset including $\emptyset$ and $N$ is an offer set), and rho j i is $\rho_{j,i}$, the transition from $j$ to $i$. The model structure carries the paper's standing assumptions $\lambda_j>0$ and $\sum_i\rho_{j,i}<1$ (p. 1325) and the implicit nonnegativity $\rho_{j,i}\ge0$. Revenues are arbitrary reals with no sign assumption. $\mathbb R^{2n}$ is (Fin n → ℝ) × (Fin n → ℝ), and extreme points are Mathlib's Set.extremePoints ℝ.

The pair $(P_S,R_S)$ is a solution of (Balance) chosen by Classical.epsilon; milestone 1 states that it satisfies (Balance), is nonnegative and equals every solution, so all statements are about the paper's $(P_S,R_S)$ and not about a junk value. Every optimum (of (Assortment), of the linear program over $\mathcal H$, of (Dual)) is stated as "feasible and at least as good as every feasible point", never through sSup or sInf. The goal is stated for every optimal $\hat v$ of (Dual), which is what the proof uses; uniqueness of $\hat v$ is neither assumed nor claimed. Replacing the hypothesis "$\hat v$ optimal for (Dual)" by "$\hat v$ feasible for (Dual)" would make the statement false, and an existential over $\hat v$ would weaken it; neither is the target.

No hypothesis beyond the paper's is added. Two printed index slips on pp. 1325–1326 (a garbled sum in the discussion after (Balance), and $\sum_i\rho_{i,j}\hat z_j$ for $\sum_i\rho_{i,j}\hat z_i$ in the proof of Lemma 1) are not part of any statement here.

Welcome contributions: the existence and uniqueness of (Balance) solutions via Neumann series for substochastic matrices, a characterization of vertices of polyhedra given by equality constraints and nonnegativity, and a strong duality statement usable for this primal–dual pair. These are reusable well beyond this mission.

Selected references

• J. B. Feldman, H. Topaloglu, Revenue Management Under the Markov Chain Choice Model, Operations Research 65(5):1322–1342, 2017. https://doi.org/10.1287/opre.2017.1628
• J. Blanchet, G. Gallego, V. Goyal, A Markov Chain Approximation to Choice Modeling, Operations Research 64(4):886–905, 2016. https://doi.org/10.1287/opre.2016.1505
• M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994. https://doi.org/10.1002/9780470316887

Motivation

Chunk 10 formalized the discrete conjugacy theorem — the Legendre-Fenchel transform's bijection between the classes of M-convex and L-convex functions — and, along the way, a function-level generalization of Edmonds's intersection theorem. Section 8.4 turns that machinery toward a different question: not "how are two convexity classes related," but "when does a discrete optimization problem have a dual that meets it with equality." The classical route to such strong-duality results in continuous convex programming — Lagrangian relaxation, an embedding of the problem in a family of perturbed problems, and a saddle-point characterization of when primal and dual values coincide — has a discrete analogue that needs no continuity, no differentiability, and no convexity in the classical sense at all: only the elementary fact that the Legendre-Fenchel transform, once discretized, is still an involution on the right class of functions. This mission formalizes that discrete Lagrangian duality framework and its central saddle-point theorem, in full generality — before the book specializes it, in the section that follows, to the specific M-convex perturbation that gives the chapter's headline strong-duality result for M-convex programs.

Setting

Let $V$ and $U$ be finite ground sets. A perturbation of an optimization problem $\min\{f(x) : x \in \mathbb Z^V\}$ is a function $F : \mathbb Z^V \times \mathbb Z^U \to \mathbb Z \cup \{+\infty\}$ such that $F(x,0) = f(x)$ for all $x$ (Eq. (8.54)) and, for each fixed $x$, $F(x,\cdot)$ is self-biconjugate: $F(x,\cdot)^{\bullet\bullet} = F(x,\cdot)$ under the discrete Legendre-Fenchel transform of chunk 10 (Eq. (8.55)). The Lagrangian function is $K(x,y) = \inf\{F(x,u) + \langle u,y\rangle : u \in \mathbb Z^U\}$ (Eq. (8.58)), valued in $\mathbb Z \cup \{\pm\infty\}$ (formalized in EReal, since both the infimum and the supremum below can be genuinely unbounded). The dual objective is $g(y) = \inf\{K(x,y) : x \in \mathbb Z^V\}$ (Eq. (8.60)). Writing $\inf(P) = \inf_x f(x)$, $\sup(D) = \sup_y g(y)$, $\operatorname{opt}(P) = \{x : f(x) = \inf(P)\}$, $\operatorname{opt}(D) = \{y : g(y) = \sup(D)\}$, the primal problem $P$ is to minimize $f$ over $\mathbb Z^V$ and the dual problem $D$ is to maximize $g$ over $\mathbb Z^U$.

Formalization targets

Goal: Theorem 8.54 (the saddle-point theorem)

Assuming $F$ is self-biconjugate (Eq. (8.55)): both $\inf(P)$ and $\sup(D)$ are finite and $\min(P) = \max(D)$ if and only if there exist $\bar x \in \mathbb Z^V$, $\bar y \in \mathbb Z^U$ with $K(\bar x,\bar y)$ finite and $K(x,\bar y) \le K(\bar x,\bar y) \le K(\bar x,y)$ for all $x,y$ — a saddle point of the Lagrangian kernel. When this holds, $\bar x \in \operatorname{opt}(P)$ and $\bar y \in \operatorname{opt}(D)$.

Milestones: Theorem 8.52, Proposition 8.51(1)-(2)

Theorem 8.52 (weak duality): $\inf(P) \ge \sup(D)$ always, with no biconjugacy hypothesis on $F$ at all — the baseline the saddle-point theorem sharpens to equality. Proposition 8.51(1)-(2): under self-biconjugacy, the perturbation $F$ (and hence the primal objective $f$) is itself recoverable from the Lagrangian kernel $K$ by a supremum, $F(x,u) = \sup_y\{K(x,y) - \langle u,y\rangle\}$ and $f(x) = \sup_y K(x,y)$ — the algebraic identity the saddle-point theorem's proof turns on directly.

Significance

The result itself. The saddle-point theorem is the general-purpose engine behind every strong-duality result the book proves for specific classes of discrete optimization problems: the book's own next section specializes it (via a particular choice of $F$ built from an M-convex regularizer $r$) to obtain strong duality for M-convex programs, but the theorem itself needs no M-convexity, no submodularity, and no exchange axiom — only the elementary self-biconjugacy of a perturbation under the discrete Legendre-Fenchel transform. It is, in that sense, the most general and most reusable strong-duality statement in the book: any future mission proving strong duality for a specific class of discrete programs (M-convex, M2-convex, network flow, or otherwise) by exhibiting a self-biconjugate perturbation can cite this theorem directly rather than reproving the saddle-point argument from scratch.

Formalizing it. No matching item exists on the platform for a discrete Lagrangian saddle- point theorem, discrete weak duality, or this perturbation-based duality framework. (A prior-art search turned up an unrelated continuous Lagrangian saddle-point theorem for convex cones, Luenberger's Chapter 8 §8.4, formalized as VectorSpaceOpt.lagrangian_saddle_sufficient_pointed — a genuinely different setting: no discreteness, no biconjugacy hypothesis, and a one-directional sufficiency statement rather than this mission's iff. Not reused.) This mission gives the first formal statement of discrete Lagrangian duality, and directly reuses chunk 10's ConvexConjugate apparatus (self-biconjugacy is stated using chunk 10's own conjugate-of-conjugate composition), demonstrating exactly the kind of shared-substrate payoff the discrete conjugacy theorem was built to provide.

Difficulty

The saddle-point theorem's "only if" direction is not a routine unwinding of definitions: given $\min(P) = \max(D)$ at finite common value, one must construct the saddle point $(\bar x,\bar y)$ — the book's proof takes $\bar x \in \operatorname{opt}(P)$, $\bar y \in \operatorname{opt}(D)$ (which exist because the infimum/supremum are attained at a finite optimum) and verifies the sandwiching inequality using Proposition 8.51(2)'s identity $f(x) = \sup_y K(x,y)$ together with weak duality, rather than by any direct algebraic manipulation of $K$ alone. Skipping straight to a "trivial" biconditional that never invokes Proposition 8.51 would misrepresent the actual proof structure the book relies on for exactly this direction.

Formalization scope

$V, U$ are Fintype ground types; LagrangianKernel, DualObjective, InfP, SupD are EReal-valued to keep both the defining infima/suprema total (a complete lattice) without an artificial finiteness side-condition; OptP, OptD compare PrimalValue/DualObjective against InfP/SupD after casting through chunk 10's ToEReal, mirroring that chunk's own round-trip convention. "Finite" throughout is formalized as ≠ ⊤ ∧ ≠ ⊥ in EReal. The perturbation $F$ itself is left fully abstract (an arbitrary function satisfying the self-biconjugacy hypothesis where needed) — this mission does not draft the specific M-convex perturbation $F_r$ (Eq. (8.61)) that the book's next subsection (§8.4.3) uses to specialize this framework to M-convex programs, nor Theorem 8.59 (the resulting M-convex strong-duality theorem) itself, which needs that specific perturbation plus its own regularity conditions (REG)/(OBJ) and a chain of M-convex-specific propositions (8.55–8.58) beyond what the general framework built here provides. A trivializing formalization would state the saddle-point theorem's sandwiching inequality with a weaker order (e.g., only one of the two directions) or would omit the "$\bar x \in \operatorname{opt}(P), \bar y \in \operatorname{opt}(D)$" consequence clause; neither is done — both inequalities and the full consequence clause are included exactly as the book states them.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.

Motivation

Airlines, hotels and rental firms sell a fixed stock of capacity (seats on a flight leg, rooms on a night) over a finite selling horizon, and control revenue mainly by deciding which products (fare classes, rate plans) to make available at each moment. When customers substitute between products, the decision in each period is an assortment: a subset of products to offer. Talluri and van Ryzin (Management Science 2004) formulated this single resource revenue management problem as a dynamic program for a general choice model, and showed that structural properties of the optimal policy depend strongly on how customers choose.

The Markov chain choice model of Blanchet, Gallego and Goyal (technical report 2013; Oper. Res. 2016) describes substitution by a Markov chain on the products and approximates a broad class of random-utility models. Feldman and Topaloglu (Oper. Res. 65(5), 2017) study assortment and revenue management problems under this model. This mission formalizes their structural result for the single resource problem (Theorem 5): there is an optimal policy whose offer sets are nested in the remaining capacity and in the remaining time, so that it can be implemented by protection levels, one capacity threshold per product and period.

Setting

There are $n$ products $N=\{1,\dots,n\}$. A customer arrives to purchase product $j$ with probability $\lambda_j$. If $j$ is offered she buys it; otherwise she moves to product $i$ with probability $\rho_{j,i}$, and leaves with probability $1-\sum_{i\in N}\rho_{j,i}$. The paper assumes $\lambda_j>0$ and $\sum_{i\in N}\rho_{j,i}<1$ for all $j$. For an offer set $S\subseteq N$, the purchase probabilities $P_{j,S}$ and the visit counts $R_{j,S}$ of unavailable products solve the (Balance) equations

$$P_{j,S}+R_{j,S}=\lambda_j+\sum_{i\in N}\rho_{i,j}R_{i,S}\ \ \forall j,\qquad P_{j,S}=0\ \ (j\notin S),\qquad R_{j,S}=0\ \ (j\in S).$$

With product revenues $r_j$, the (Assortment) problem is $\max_{S\subseteq N}\sum_{j}P_{j,S}r_j$, and its (Dual) linear program is

$$\min_{v\in\mathbb R^n}\Big\{\sum_{j}\lambda_jv_j : v_j\ge r_j,\ v_j\ge\sum_{i}\rho_{j,i}v_i\ \ \forall j\Big\}.$$

In the single resource problem there are $T$ periods and $c$ units of capacity. In each period at most one customer arrives; a sale of product $j$ earns $r_j$ and uses one unit. The optimal expected revenue $V_t(x)$ from period $t$ on with $x$ units left satisfies the (Single Resource) dynamic program

$$V_t(x)=\max_{S\subseteq N}\Big\{\sum_{j\in N}P_{j,S}\{r_j+V_{t+1}(x-1)-V_{t+1}(x)\}\Big\}+V_{t+1}(x),$$

with $V_{T+1}(x)=0$ and $V_t(0)=0$. An optimal subset $\hat S_t(x)$ is a maximizer of the problem on the right side. The marginal value of capacity is $\Delta V_t(x)=V_t(x)-V_t(x-1)$.

Formalization targets

Goal: Theorem 5 (p. 1329)

There exists an optimal policy, i.e. a choice of maximizers $\hat S_t(x)$ for all $1\le t\le T$, $1\le x\le c$, such that

$$\hat S_t(x-1)\subseteq \hat S_t(x)\qquad\text{and}\qquad \hat S_{t-1}(x)\subseteq\hat S_t(x).$$

The offer set shrinks as capacity runs down, and it is smaller when more periods remain.

Milestones

1. (§2, p. 1325) The (Balance) equations have a unique nonnegative solution for every $S$.
2. (Theorem 2, p. 1326) If $\hat v$ is optimal for (Dual), then $\{j:\hat v_j=r_j\}$ is optimal for (Assortment).
3. (Lemma 3, p. 1327) For $\eta\ge0$, with $\hat v^\eta$ optimal for (Dual) with revenues $r_j-\eta$,

$$\{j:\hat v^\eta_j=r_j-\eta\}\subseteq\{j:\hat v^0_j=r_j\}.$$

4. ((Single Resource), pp. 1328–1329) The value functions satisfy the printed recursion $V_t(x)=\max_S\{\sum_jP_{j,S}\{r_j+V_{t+1}(x-1)\}+\{1-\sum_jP_{j,S}\}V_{t+1}(x)\}$ and the boundary conditions.
5. (Proof of Theorem 5, p. 1329) $\Delta V_{t+1}(x)\le\Delta V_{t+1}(x-1)$ and $\Delta V_{t+1}(x)\le\Delta V_t(x)$.

Significance

Theorem 5 makes the optimal policy a protection level policy: for each product $j$ and period $t$ there is a threshold $\bar x_{jt}$ such that $j$ is offered exactly when at least $\bar x_{jt}$ units remain. The policy can then be stored as $n\,T$ numbers instead of a table of subsets, and each product can be controlled separately, which is how airline inventory systems are organized. The paper also shows (Table 2, p. 1330) that the products need not be closed in revenue order: a higher-fare product can be closed before a lower-fare one, so the result is genuinely about nested sets, not about nested fare classes. Talluri and van Ryzin (2004) show that under the multinomial logit model an optimal assortment consists of a number of products with the largest revenues; the Markov chain choice model does not have this property (Table 1, p. 1328), so their structure of the optimal policy does not carry over directly.

The result is proved in the paper. No part of it is formalized on Prove2Me. The monotonicity of marginal values for an abstract choice model is published and proved as RevenueManagement.choice_marginal_values, stated over the definitions RevenueManagement_singleResource; milestone 5 is the same statement for this paper's dynamic program and can be bridged to it. A complete development here produces machine-checked Theorem 2 and Lemma 3 for the Markov chain choice model, which are reusable for any assortment problem under this model.

Difficulty

The obvious argument chooses, for each $(t,x)$, any maximizer of the stage problem. This fails: the stage problems have ties, and an arbitrary choice of maximizers need not be nested, so the theorem is an existence statement about a coordinated choice. The dependence of $P_{j,S}$ on $S$ is through the solution of a linear system, so the effect of adding or removing a product on the other purchase probabilities has no simple sign, and optimal assortments need not be nested by revenue (Table 1, p. 1328). The link between two stage problems is that their revenues differ by the same constant for all products, and what has to be shown is that such a uniform shift moves an optimal assortment in a controlled direction. Revenues in the stage problems, $r_j-\Delta V_{t+1}(x)$, can be negative.

Formalization scope

Products are Fin n, offer sets Finset (Fin n), the paper's $\subset$ is non-strict inclusion $\subseteq$. The model is a structure with fields $\lambda$, $\rho$ and the standing assumptions $\lambda_j>0$, $\sum_i\rho_{j,i}<1$; $\rho_{j,i}\ge0$ is added as a field because the $\rho_{j,i}$ are probabilities. $(P_S,R_S)$ is a solution of (Balance) chosen by Classical.epsilon; milestone 1 states that it is the unique nonnegative one. The value function is defined by recursion on the number of periods to go, and $V_t$ is used only for $1\le t\le T+1$. Maxima over $S\subseteq N$ are taken over the finite family of all subsets, so they are attained; dual optimality is "feasible and no worse than every feasible point".

One hypothesis is added to Theorem 5 and milestone 5, and disclosed: $\sum_j\lambda_j\le1$, implied by the paper's description of at most one arrival per period (it makes $1-\sum_jP_{j,S}$ a probability). No sign condition is placed on the revenues, as on the page. Theorem 2 and Lemma 3 are stated for arbitrary real revenues, as the proof of Theorem 5 applies them to $r_j-\Delta V_{t+1}(x)$. The first inclusion of Theorem 5 is stated for $x\ge2$: with $x-1=0$ units there is no decision, and the paper reads $\hat S_t(0)$ as $\emptyset$. The existential requires every $\hat S_t(x)$ in range to be optimal for its stage problem; a statement without that clause would be satisfied by the empty sets and is ruled out.

Theorem 2 and Lemma 3 need LP duality for (Dual) and the (Balance) system; milestone 5 needs the standard induction on the dynamic program, or a bridge to the published choice_marginal_values (with arrival probabilities $1$ and choice model $P_{j,S}$). Proofs of any milestone, and bridges to published Mathlib or platform LP duality results, are welcome.

Selected references

• J. B. Feldman, H. Topaloglu, Revenue Management Under the Markov Chain Choice Model, Operations Research 65(5):1322–1342, 2017. https://doi.org/10.1287/opre.2017.1628
• J. Blanchet, G. Gallego, V. Goyal, A Markov Chain Approximation to Choice Modeling, Operations Research 64(4):886–905, 2016. https://doi.org/10.1287/opre.2016.1505
• K. Talluri, G. van Ryzin, Revenue Management Under a General Discrete Choice Model of Consumer Behavior, Management Science 50(1):15–33, 2004. https://doi.org/10.1287/mnsc.1030.0147
• K. Talluri, G. van Ryzin, The Theory and Practice of Revenue Management, Springer, 2004. https://doi.org/10.1007/b139000

Motivation

Network revenue management decides which products to make available to arriving customers when products share scarce resources: an airline sells itineraries (products) that consume seats on flight legs (resources), and each itinerary has a fare. When customers choose among the offered products, rather than asking for one fixed product, the standard planning tool is a deterministic linear program that replaces random choices by their expected values. Gallego, Iyengar, Phillips and Dubey (2004, Columbia CORC technical report TR-2004-01) and Liu and van Ryzin (2008) formulated this choice-based linear program; its solution drives bid-price and offer-set policies used in practice.

The difficulty is size. The choice-based program has one variable for each subset of products, $2^n$ in all, and is solved by column generation, whose pricing subproblem is itself an assortment problem. Feldman and Topaloglu (Oper. Res. 65(5), 2017) show that when customers choose under the Markov chain choice model of Blanchet, Gallego and Goyal (2016), the choice-based program is equivalent to a linear program with only $2n$ variables and $m+n$ constraints. This mission formalizes that equivalence, Theorem 7 of the paper.

Setting

There are $n$ products, $N=\{1,\dots,n\}$. Under the Markov chain choice model a customer first visits product $j$ with probability $\lambda_j$. If the product she visits is offered, she buys it. Otherwise she moves from product $j$ to product $i$ with probability $\rho_{j,i}$, or leaves without buying with probability $1-\sum_{i\in N}\rho_{j,i}$. The paper assumes throughout that

$$\lambda_j>0\quad\text{and}\quad\sum_{i\in N}\rho_{j,i}<1\qquad\text{for all } j\in N.$$

For an offer set $S\subseteq N$, $P_{j,S}$ is the expected number of visits to product $j$ while it is offered, which is its purchase probability, and $R_{j,S}$ is the expected number of visits to $j$ while it is not offered. The pair $(P_S,R_S)$ is the solution of the (Balance) equations

$$P_{j,S}+R_{j,S}=\lambda_j+\sum_{i\in N}\rho_{i,j}R_{i,S}\ \ \forall j\in N,\qquad P_{j,S}=0\ \ \forall j\notin S,\qquad R_{j,S}=0\ \ \forall j\in S .$$

Dropping the constraints tied to $S$ gives the polyhedron

$$\mathcal H=\Big\{(x,z)\in\mathbb R^{2n}_+ : x_j+z_j=\lambda_j+\sum_{i\in N}\rho_{i,j}z_i\ \ \forall j\in N\Big\}.$$

The network has $m$ resources, $M=\{1,\dots,m\}$, with capacities $c_q$; the selling horizon has $T$ periods; product $j$ earns $r_j$ and consumes $a_{q,j}$ units of resource $q$. With $u_S$ the probability of offering $S$ in a period, the (Choice Based) linear program is

$$\max_{u\in\mathbb R^{2^n}_+}\Big\{\sum_{S\subseteq N}\sum_{j\in N}T r_jP_{j,S}u_S:\ \sum_{S\subseteq N}\sum_{j\in N}Ta_{q,j}P_{j,S}u_S\le c_q\ \forall q\in M,\ \sum_{S\subseteq N}u_S=1\Big\},$$

and the (Reduced) linear program is

$$\max_{(x,z)\in\mathbb R^{2n}_+}\Big\{\sum_{j\in N}T r_jx_j:\ \sum_{j\in N}Ta_{q,j}x_j\le c_q\ \forall q\in M,\ x_j+z_j=\lambda_j+\sum_{i\in N}\rho_{i,j}z_i\ \forall j\in N\Big\}.$$

In (Reduced), $x_j$ is the expected number of visits to product $j$ while it is available and $z_j$ the expected number of visits while it is not.

Formalization targets

Goal: Theorem 7

Let $(\hat x,\hat z)$ be an optimal solution of (Reduced). Then there are subsets $S^1,\dots,S^K\subseteq N$ and positive scalars $\gamma^1,\dots,\gamma^K$ with $\sum_k\gamma^k=1$ such that

$$\hat x=\sum_{k=1}^K\gamma^kP_{S^k},\qquad \hat z=\sum_{k=1}^K\gamma^kR_{S^k},$$

and for any such subsets and scalars the vector $\hat u$ with $\hat u_{S^k}=\gamma^k$ and $\hat u_S=0$ for $S\notin\{S^1,\dots,S^K\}$ is optimal for (Choice Based), with objective value equal to that of $(\hat x,\hat z)$ in (Reduced). In particular the two programs have the same optimal value.

Milestones

1. (Balance) has a unique and nonnegative solution for every offer set (§2, p. 1325).
2. Lemma 1: an extreme point $(\hat x,\hat z)$ of $\mathcal H$ equals $(P_{S},R_{S})$ for $S=\{j:\hat x_j>0\}$ (p. 1326).
3. Lemma 10: $\mathcal H$ is bounded (quoted on p. 1331; proved in the online appendix).
4. Every point of $\mathcal H$ is a positive convex combination of finitely many extreme points of $\mathcal H$ (proof of Theorem 7, p. 1331).
5. Every feasible $u$ of (Choice Based) yields the feasible point $\tilde x_j=\sum_S P_{j,S}u_S$, $\tilde z_j=\sum_S R_{j,S}u_S$ of (Reduced), with the same objective value (proof of Theorem 7, p. 1332).

Significance

The result. Theorem 7 replaces a program with $2^n$ columns by one with $2n$ variables and $m+n$ constraints, solvable directly by any LP solver, and it returns an optimal solution of the original program, not only its value. The optimal value is the standard upper bound on the optimal expected revenue of a network revenue management policy, and the dual variables of the capacity constraints are the bid prices used to control sales. The decomposition of part 1 is what turns the small program's solution back into offer-set frequencies that a policy can implement; Section 7 of the paper makes that decomposition algorithmic (a separate mission of this series).

Formalizing it. The theorem is proved in the paper; to the best of available knowledge no machine-checked version exists. A formal proof checks the link between polyhedral geometry (extreme points of $\mathcal H$ and the solutions of (Balance)) and linear-programming optimality, and records exactly which properties of the Markov chain choice model are used: the standing assumptions enter through uniqueness and nonnegativity of $(P_S,R_S)$ and through boundedness of $\mathcal H$.

Difficulty

The inequality "(Reduced) $\ge$ (Choice Based)" is a direct computation: averaging the (Balance) equations with weights $u_S$ lands in $\mathcal H$. The reverse direction is where the obvious argument fails. A point of $\mathcal H$ has no offer set attached to it, and a general polyhedron need not be the convex hull of its extreme points: it can contain lines or rays. The argument requires that $\mathcal H$ is bounded, which depends on the substochasticity $\sum_i\rho_{j,i}<1$, and that each extreme point is exactly some $(P_S,R_S)$, which uses the structure of the balance equations and the uniqueness of their solution. Neither follows from general linear-programming facts. In Lean, the finite vertex representation of a bounded polyhedron is also not a one-line consequence of Mathlib's Krein–Milman theorem, which gives only the closure of the convex hull.

Formalization scope

Products are Fin n, offer sets Finset (Fin n), resources Fin m. The model is a structure Model n holding $\lambda$, $\rho$ (rho j i $=\rho_{j,i}$, the transition from $j$ to $i$) and the standing assumptions $\lambda_j>0$ and $\sum_i\rho_{j,i}<1$; the nonnegativity $\rho_{j,i}\ge0$, implicit in the paper because the $\rho_{j,i}$ are probabilities, is an added field. $(P_S,R_S)$ is a solution of (Balance) chosen by Classical.epsilon, and milestone 1 is what identifies it with the paper's unique solution. $\mathcal H$ is a subset of (Fin n → ℝ) × (Fin n → ℝ), extreme points are Mathlib's Set.extremePoints ℝ, and boundedness is Bornology.IsBounded. $T$ is a natural number entering as a real factor exactly where the paper writes it; $c$, $a$, $r$ carry no sign conditions, as in the paper. "Optimal solution" means feasible and at least as good as every feasible point; no supremum is used.

Two conventions are disclosed. The paper defines $\hat u$ by $\hat u_{S^k}=\gamma^k$; the formalization sets $\hat u_S=\sum_{k:S^k=S}\gamma^k$, which agrees when the $S^k$ are distinct and is the only consistent reading otherwise. Milestone 5 is stated for every feasible $u$ of (Choice Based), while the paper applies it to an optimal one; its argument uses only feasibility. No printed statement needed correction.

The goal is not only the decomposition of part 1, which is milestones 2 and 4 combined: it also asserts optimality of $\hat u$ and equality of the optimal values, and a formalization that drops part 2 does not state Theorem 7. Part 2 is required for every decomposition, and part 1 guarantees one exists, so part 2 is not vacuous.

A complete development needs the vertex representation of polytopes (reusable beyond this mission; LinearOptimization.polyhedron_resolution on the platform proves the resolution theorem in another encoding), the theory of substochastic matrices behind (Balance) (invertibility of $I-\bar Q$ with a nonnegative inverse), and finite-sum manipulations over Finset (Fin n). Contributions to any milestone, and bridges to existing polyhedral results, are welcome.

Selected references

• J. B. Feldman, H. Topaloglu, Revenue Management Under the Markov Chain Choice Model, Operations Research 65(5):1322–1342, 2017. https://doi.org/10.1287/opre.2017.1628
• J. Blanchet, G. Gallego, V. Goyal, A Markov Chain Approximation to Choice Modeling, Operations Research 64(4):886–905, 2016. https://doi.org/10.1287/opre.2016.1505
• Q. Liu, G. van Ryzin, On the Choice-Based Linear Programming Model for Network Revenue Management, Manufacturing & Service Operations Management 10(2):288–310, 2008. https://doi.org/10.1287/msom.1070.0172
• G. Gallego, G. Iyengar, R. Phillips, A. Dubey, Managing Flexible Products on a Network, Computational Optimization Research Center Technical Report TR-2004-01, Columbia University, 2004 (technical report; no DOI).
• M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994. https://doi.org/10.1002/9780470316887