Decision Rule-Based Valid Inequalities

• Decision rule-based valid inequalities are mathematical expressions derived from decision rules that convert qualitative admissibility conditions into verifiable constraints.
• They appear across diverse fields such as social choice, online learning, mixed-integer optimization, and percolation, providing guarantees like transitivity and anytime-valid error control.
• These inequalities enhance computational efficiency by tightening relaxations, reducing branch-and-bound nodes, and ensuring valid inference under adaptive decision-making.

Decision rule-based valid inequalities, in the body of work represented here, are inequalities or equations whose validity is certified by an explicit rule for aggregating preferences, selecting splits, choosing disjunctive terms, or adaptively revealing information. The underlying rule may be a majority decision rule, a sequential stopping rule, a split disjunction in a decomposition algorithm, a branch-and-bound partition, or a decision-tree exploration of a probabilistic configuration. Across these settings, the common mathematical pattern is the conversion of a qualitative admissibility condition into a checkable inequality, equation, or cut-generating program (Hou, 2022, Amoukou et al., 29 May 2026, Fang et al., 4 Jun 2025, Towle et al., 2019, 2207.13619, Gladkov, 2024).

1. Terminological scope and structural idea

The cited literature does not use a single canonical definition of “decision rule-based valid inequalities,” but it does exhibit a recurring structure. In social choice, Sen’s value restriction is translated into inequalities and equations using preference maps and membership functions. In online learning, split commitment is turned into an anytime-valid testing rule with thresholds defined by a wealth process or a confidence sequence. In mixed-integer optimization, disjunctive programming and Benders Decomposition use split-selection rules, cut-generating LPs, and point-ray constructions to derive inequalities valid for convex hulls. In percolation, decision trees induce hybrid configurations that generalize classical correlation and disjoint-occurrence inequalities (Hou, 2022, Amoukou et al., 29 May 2026, Fang et al., 4 Jun 2025, 2207.13619, Gladkov, 2024).

This suggests a broad interpretation in which the “decision rule” is the mechanism that specifies which configurations are compared, aggregated, or excluded, while the “valid inequality” is the algebraic object that encodes the consequence of that rule. The rule may be deterministic, adaptive, randomized, or disjunctive; the validity claim may concern a polyhedral hull, a transitivity condition, a sequential error rate, or a probability bound.

A further implication of the cited works is that validity is not confined to cutting planes in the narrow integer-programming sense. Some valid inequalities are polyhedral cuts for a relaxation, some are structural inequalities for preference profiles, and some are probabilistic inequalities for events in product spaces. The term therefore spans several technical traditions rather than a single solver-centric methodology.

2. Preference maps and majority-rule transitivity

A precise non-polyhedral instance appears in the description of Sen’s transitivity condition for the majority decision rule. For a triple of alternatives $(x,y,z)$, an individual’s weak preference ordering is encoded as a preference map (PM), whose entries are sets containing the ranking position or positions assigned to each alternative. If a voter ranks $x>y>z$, then the PM has $x:\{1\}$, $y:\{2\}$, $z:\{3\}$; if $x \sim y > z$, then $x:\{1,2\}$, $y:\{1,2\}$, $z:\{3\}$ (Hou, 2022).

The paper reformulates Sen’s value restriction for each triple $t$ by considering the set $x>y>z$0 of “concerned” individuals and taking, for each alternative $x>y>z$1, the union of positions that this alternative occupies across the PMs of individuals in $x>y>z$2. The condition is then written as inequality (3): $x>y>z$3 The paper interprets this as the existence, for each triple, of at least one alternative for which some ranking position is never assigned among the concerned individuals, so that the union of attainable positions has cardinality less than $x>y>z$4. In the terminology of the summary, this “blocks” the formation of a cycle in the majority outcome and yields a transitive social preference under majority rule (Hou, 2022).

The same condition is also converted into an equation by means of the membership preference map (mPM), a $x>y>z$5 binary matrix in which

$x>y>z$6

if alternative $x>y>z$7 can take position $x>y>z$8 in individual $x>y>z$9’s PM, and $x:\{1\}$0 otherwise. Sen’s condition is then expressed as equation (4): $x:\{1\}$1 Thus, for each triple, there exists an alternative-position pair that no concerned individual assigns. The paper presents these formulations as checkable criteria, and explicitly notes that the PM-based union-cardinality test and the mPM-based zero-cell condition make Sen’s value restriction concrete and computable (Hou, 2022).

The significance of this construction is methodological. A qualitative condition on ordinal preferences is replaced by algebraic conditions on set unions, cardinalities, and membership functions. This is a decision rule-based validity statement in the sense that the majority decision rule is the governing rule, while the valid inequalities and equations encode the exact structural restriction needed for transitivity.

3. Sequential split decision rules in online decision trees

A statistically different but structurally related use of decision rules appears in online decision trees. Hoeffding Trees test whether a candidate split is significantly better than its alternatives using fixed-sample concentration inequalities. The cited work argues that this practice is formally invalid because split decisions are made at a data-dependent stopping time; fixed-sample guarantees control error only at a predetermined sample size, and under sequential monitoring the probability of incorrect splits can become much larger than anticipated, even approaching $x:\{1\}$2 (Amoukou et al., 29 May 2026).

The proposed alternative formulates split selection as an online hypothesis-testing problem comparing an incumbent unsplit predictor with a challenger induced by a candidate split. For a candidate split $x:\{1\}$3 at node $x:\{1\}$4, the instantaneous improvement is

$x:\{1\}$5

The first decision rule is a betting-based test under the strong null

$x:\{1\}$6

with wealth process

$x:\{1\}$7

By Ville’s inequality,

$x:\{1\}$8

so the split is committed at

$x:\{1\}$9

The paper states that, with a suitable allocation of the $y:\{2\}$0, false splits are globally controlled across the tree’s lifetime (Amoukou et al., 29 May 2026).

The second rule is confidence-sequence based and tests a weak null on the average advantage: $y:\{2\}$1 A time-uniform empirical Bernstein confidence sequence is used, with radius

$y:\{2\}$2

and lower bound $y:\{2\}$3. The split is committed when $y:\{2\}$4 (Amoukou et al., 29 May 2026).

The formal guarantees are explicitly theorem-level. Theorem 1 gives anytime-valid global control: $y:\{2\}$5 Theorem 2 states finite commitment time under a persistent advantage, with expected stopping time scaling as $y:\{2\}$6. Theorem 3 states that under stationary i.i.d. data and convex loss, expected risk decreases monotonically as the tree grows using anytime-valid splitting, both between and at split commits (Amoukou et al., 29 May 2026).

Here the “valid inequality” is not a polyhedral cut but a time-uniform bound that remains valid under optional stopping. The decision rule is the split-commitment rule itself, defined by threshold crossing of $y:\{2\}$7 or positivity of the CS lower bound. The paper also reports empirical improvement for standalone trees and Adaptive Random Forests on non-stationary streams, together with substantially smaller trees (Amoukou et al., 29 May 2026).

4. Disjunctive Benders cuts and split-selection rules

In mixed-integer optimization, the most direct realization of decision rule-based valid inequalities arises from disjunctive programming. Conventional Benders Decomposition generates cuts that are typically tight only for the continuous relaxation. The cited work proposes Disjunctive Benders Decomposition, which generates valid inequalities for the convex hull of the Benders reformulation by integrating disjunctive programming theory with Benders Decomposition (Fang et al., 4 Jun 2025).

The central rule is an integral split disjunction

$y:\{2\}$8

with corresponding sides

$y:\{2\}$9

Valid inequalities for $z:\{3\}$0 are characterized through a cut-generating LP, and the tightest separating cut at $z:\{3\}$1 is obtained from

$z:\{3\}$2

This yields supporting hyperplanes for the convex hull of the disjunctive union (Fang et al., 4 Jun 2025).

Because $z:\{3\}$3 depends on exponentially many dual extreme points and rays, the paper does not solve the full CGLP directly. Instead, Routine 1 maintains a relaxation, projects the current solution to each side of the split, and calls a typical Benders oracle to separate the point from $z:\{3\}$4. Violated cuts are then added in lifted form to both sides via the “raising” process described in Proposition 2.4. The rule-based aspect is explicit in split selection: for mixed-binary problems, splits usually correspond to variables with fractional values; in the specialized strategy, one splits lexicographically on the highest-index fractional variable and reuses previously found disjunctive cuts in later splits (Fang et al., 4 Jun 2025).

Theoretical consequences are substantial. For mixed-binary problems, Theorem 2.2 states that there is no need to solve the master problem as a mixed-integer program; the strengthened continuous master together with the disjunctive cuts suffices for finite convergence. Computationally, the method is reported to reduce branch-and-bound nodes by a factor of $z:\{3\}$5 or more on large facility location instances, and by over an order of magnitude on difficult stochastic network interdiction instances, while mitigating the tail-off effect of conventional Benders Decomposition (Fang et al., 4 Jun 2025).

This setting exhibits the classical optimization meaning of a decision rule-based valid inequality. A split rule determines a disjunction, the disjunction induces a hull, and the cut-generating routine returns inequalities valid for that hull rather than merely for the continuous relaxation.

5. Reverse convex sets, external intersection disjunctions, and point-ray formulations

A related development concerns reverse convex sets of the form $z:\{3\}$6, where $z:\{3\}$7 is a polyhedron and $z:\{3\}$8 is an open convex set known not to contain feasible solutions. Classical intersection cuts, following Balas, are generated from a basic solution $z:\{3\}$9 of $x \sim y > z$0 lying inside $x \sim y > z$1. The cited reverse-convex framework extends this by deriving valid inequalities from basic solutions $x \sim y > z$2 (Towle et al., 2019).

For the cone $x \sim y > z$3 at an external basis, the rays are partitioned into those that never enter $x \sim y > z$4, those that enter and remain in $x \sim y > z$5, and those that enter and later exit $x \sim y > z$6. The resulting two-term intersection disjunction states that for all $x \sim y > z$7, at least one of

$x \sim y > z$8

must hold. When one side is empty over $x \sim y > z$9, the other inequality is valid for all $x:\{1,2\}$0; the summary calls these “external intersection cuts.” The paper then generalizes to a multi-term disjunction using the recession cone $x:\{1,2\}$1, and proposes a family of inequalities of the form

$x:\{1,2\}$2

The paper explicitly notes that the systematic choice of subsets $x:\{1,2\}$3 is conceptually related to “decision rules,” since the cut family is indexed by combinatorial decisions on which rays are active (Towle et al., 2019).

The $x:\{1,2\}$4-polyhedral disjunctive-cut framework addresses a similar issue from a different angle. Starting from a general disjunction

$x:\{1,2\}$5

the method collects, for each term $x:\{1,2\}$6, an LP-optimal point $x:\{1,2\}$7 and the basis rays $x:\{1,2\}$8 at $x:\{1,2\}$9. It then solves the point-ray linear program

$y:\{1,2\}$0

Every feasible $y:\{1,2\}$1 defines a valid inequality $y:\{1,2\}$2 for the relaxation $y:\{1,2\}$3, and the resulting cuts are called $y:\{1,2\}$4-polyhedral disjunctive cuts (VPCs) (2207.13619).

The framework is designed to avoid recursion and to remain in the original variable space rather than a higher-dimensional lift-and-project or CGLP space. The paper states that, with the simple point-ray collection, the PRLP remains small enough to scale to $y:\{1,2\}$5 terms in experiments. It also reports that on $y:\{1,2\}$6 instances, VPCs plus Gomory cuts closed an average of $y:\{1,2\}$7 of the integrality gap at the root, compared to $y:\{1,2\}$8 for Gomory cuts alone, and that adding VPCs to Gurobi’s default cuts improved the average percent gap closed from $y:\{1,2\}$9 to $z:\{3\}$0. The implementation was evaluated in the open-source COIN-OR framework, and the paper reports reductions in branch-and-bound nodes and, for a subset of harder instances, reductions in solution time (2207.13619).

These two developments share a common disjunctive logic. In both, a rule partitions the feasible region into terms—by recession-aware ray structure in reverse convex sets or by explicit disjunctive leaves in a partial branch-and-bound tree—and a cut-generating construction returns inequalities valid for the resulting hull or relaxation.

6. Decision-tree inequalities in percolation and broader interpretation

Decision-tree-based valid inequalities also arise outside optimization. In Bernoulli bond percolation on a graph $z:\{3\}$1, a decision tree is a potentially randomized procedure that reveals the states of edges one by one, with future queries allowed to depend on past revelations. Given two configurations $z:\{3\}$2 and a set $z:\{3\}$3 built by the tree, the hybrid configuration is

$z:\{3\}$4

Nikita Gladkov uses this mechanism to generalize classical percolation inequalities (Gladkov, 2024).

For upward-closed events $z:\{3\}$5, the decision-tree Harris–Kleitman inequality is

$z:\{3\}$6

For increasing events $z:\{3\}$7, the decision-tree version of the van den Berg–Kesten inequality is

$z:\{3\}$8

where $z:\{3\}$9 is defined through witnesses whose intersection is constrained by the tree-induced set $t$0. The paper applies these inequalities to connection probabilities, proving for arbitrary locally finite graphs

$t$1

and for planar graphs with $t$2 on the same face,

$t$3

It also gives further inequalities for disjoint-path events and asymmetric cluster-connectivity events (Gladkov, 2024).

The broader significance of this material is conceptual. A recurrent misconception is that valid inequalities are exclusively polyhedral cuts for mixed-integer solvers. The cited literature shows a wider domain. In social choice, the valid inequalities and equations describe when majority rule yields a transitive social preference. In online learning, they supply time-uniform inferential guarantees under adaptive stopping. In optimization, they tighten relaxations or approximate convex hulls. In percolation, they control event probabilities under adaptive exploration (Hou, 2022, Amoukou et al., 29 May 2026, Fang et al., 4 Jun 2025, Towle et al., 2019, 2207.13619, Gladkov, 2024).

This suggests that the unifying mathematical content lies in the relation between a rule for revealing, partitioning, or aggregating information and an inequality that remains valid under that rule. The rule may govern preferences, samples, disjunctions, or explored edges; the validity statement may be algebraic, geometric, sequential, or probabilistic. Within that broad interpretation, decision rule-based valid inequalities form a cross-disciplinary methodology for turning rule-governed structure into explicit mathematical constraints.