Recursive McCormick in Optimization

• Recursive McCormick is a convexification method that recursively factorizes high-order products into bilinear terms using auxiliary variables and McCormick envelopes.
• It is applied in binary polynomial optimization and multilinear programming, where the chosen recursive sequence significantly affects formulation size and bound quality.
• Recent studies show that extended flower relaxations can dominate individual recursive formulations, unifying optimal relaxation strength with efficient LP tightenings.

Recursive McCormick denotes a family of convexification and linearization techniques for multilinear and polynomial optimization in which a high-order product is recursively factorized into bilinear products, auxiliary variables are introduced for intermediate factors, and each bilinear relation is enforced by a McCormick envelope. In binary polynomial optimization, recursive McCormick relaxations are applied to the lifted multilinear set of a hypergraph; in multilinear programming, Recursive McCormick Linearization plays the analogous role for monomials over $\Omega=[0,1]^n$ or $\{0,1\}^n$. Across these settings, a central theme is that the quality and the size of the relaxation depend on the chosen recursive sequence, and recent work shows that the extended flower relaxation dominates every individual recursive McCormick relaxation while the intersection of all recursive linearizations recovers the same strength (Khajavirad, 2022, Raghunathan et al., 2022, Schutte et al., 2023).

1. Formal definitions and lifted formulations

For a hypergraph $G=(V,E)$, the binary multilinear set is

$$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$

A recursive McCormick relaxation builds a sequence of pairwise factorizations of each high-order term $z_e$ by introducing new “artificial” variables. For each original edge $e\in E$ with $|e|\ge 2$, one chooses a partition $P_e=\{J_e,K_e\}$ of $e$ into two nonempty sets, introduces $y_{J_e}$ and $\{0,1\}^n$0 if $\{0,1\}^n$1 or $\{0,1\}^n$2, marks them as new edges, and recurses until every edge in the recursion is of size $\{0,1\}^n$3 or already in $\{0,1\}^n$4. The terminal system consists of bilinear equations

$\{0,1\}^n$5

each enforced by the McCormick envelope over $\{0,1\}^n$6:

$\{0,1\}^n$7

The resulting LP in the lifted space of $\{0,1\}^n$8 is called a recursive McCormick relaxation (Khajavirad, 2022).

In the multilinear-programming formulation, the same construction is represented through triples. Let

$\{0,1\}^n$9

with $G=(V,E)$0 and $G=(V,E)$1. A subset $G=(V,E)$2 is a proper triple set if there exists $G=(V,E)$3 such that every original monomial index $G=(V,E)$4 with $G=(V,E)$5 appears as $G=(V,E)$6 for exactly one $G=(V,E)$7, and whenever a set $G=(V,E)$8 with $G=(V,E)$9 appears as a tail in some $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$0, then $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$1 also appears as $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$2 for some $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$3. Given a proper $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$4, one defines new variables $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$5 for every $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$6 that arises as a head in $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$7 and writes McCormick-envelope constraints for every triple $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$8 (Raghunathan et al., 2022).

A more general abstraction replaces recursive factorization by a directed acyclic graph $$S_G=\{(z_v,v\in V),(z_e,e\in E)\mid z_v\in\{0,1\},\ z_e=\prod_{v\in e}z_v\}.$$9 whose nodes are subsets of the base-variable set, with singletons included and with each non-singleton covered by its out-neighborhood. When every non-singleton has exactly two disjoint children, $z_e$0 is a recursive McCormick linearization. The corresponding relaxation $z_e$1 is defined by short inequalities $z_e$2, long inequalities $z_e$3, and bounds $z_e$4 (Schutte et al., 2023).

2. Recursive sequences, proper triple sets, and complexity

The recursive sequence is the chain of subsets through which an original edge is factorized down to singletons. Different sequences induce different sets of artificial variables and different McCormick constraints. The number of new variables is $z_e$5, which can grow with how finely one decomposes each edge, and finding a minimum-size recursive McCormick relaxation is NP-hard even for rank-3 hypergraphs. The effect on strength is equally significant: sequences that “reuse” the same artificial variable in multiple products can dramatically strengthen the bound, whereas if no artificial variable is shared—i.e. the relaxation is “non-overlapping”—the recursive McCormick relaxation collapses to the standard linearization by projection (Khajavirad, 2022).

The combinatorics of recursive McCormick Linearization can be encoded exactly. An exact mixed-integer programming formulation for minimum-size RML introduces binary variables

$z_e$6

where $z_e$7 if triple $z_e$8 is used anywhere and $z_e$9 if $e\in E$0 is used specifically to linearize monomial $e\in E$1. The model enforces the proper-triple conditions and minimizes $e\in E$2. In general this minimum-size problem is NP-hard; restricted to “3-MLP”, the paper shows fixed-parameter tractability with parameter $e\in E$3 desired number of triples. After kernelization to size $e\in E$4, a bounded search-tree with branching factor $e\in E$5 yields an $e\in E$6 FPT algorithm. A second exact model addresses the best-bound problem under a triple budget $e\in E$7 by formulating a bilevel program and replacing the inner LP with its dual to obtain a single-level MIP (Raghunathan et al., 2022).

These results formalize a recurring empirical observation: minimizing the number of auxiliary variables and maximizing LP strength are distinct objectives. This suggests that “small” recursive McCormick formulations and “strong” recursive McCormick formulations need not coincide.

3. Extended-flower dominance and exact strength characterizations

A standard linearization for a hypergraph $e\in E$8 introduces, for each original edge $e\in E$9 and each $|e|\ge 2$0,

$|e|\ge 2$1

Del Pia–Khajavirad introduced flower inequalities centered at an edge $|e|\ge 2$2. If $|e|\ge 2$3 and $|e|\ge 2$4 is nonempty, with $|e|\ge 2$5 for all $|e|\ge 2$6 and pairwise disjoint intersections $|e|\ge 2$7 for $|e|\ge 2$8, then

$|e|\ge 2$9

The extended flower relaxation relaxes the pairwise-disjointness condition and instead requires

$P_e=\{J_e,K_e\}$0

With the same left-hand side, these inequalities define $P_e=\{J_e,K_e\}$1, and $P_e=\{J_e,K_e\}$2 (Khajavirad, 2022).

The central theorem is that for every recursive McCormick relaxation of $P_e=\{J_e,K_e\}$3,

$P_e=\{J_e,K_e\}$4

Equivalently, the projection of any recursive McCormick formulation onto the original $P_e=\{J_e,K_e\}$5-space is implied by the extended flower relaxation. The proof proceeds by induction on the maximum depth of artificial variables. The key step is that eliminating an artificial variable by Fourier–Motzkin yields either an extended flower inequality or a weaker inequality already contained in the standard linearization (Khajavirad, 2022).

Subsequent work reformulates the same phenomenon at the level of general recursive linearizations. For a multilinear problem with hypergraph $P_e=\{J_e,K_e\}$6, let $P_e=\{J_e,K_e\}$7 denote the set of all $P_e=\{J_e,K_e\}$8 satisfying every extended flower inequality of the form

$P_e=\{J_e,K_e\}$9

where the neighbors $e$0 cover $e$1 and each intersects $e$2 non-trivially. Then $e$3 for every recursive McCormick linearization $e$4, and in fact

$e$5

The same equality holds even if one intersects over all recursive linearizations, including non-binary and non-partitioning ones (Schutte et al., 2023).

A common misconception is that any recursive factorization is automatically stronger than the standard linearization. The non-overlapping case shows otherwise, and the extended-flower results show that the maximal strength obtainable from recursive McCormick constructions is already captured by a single cut family.

4. Trilinear products, double McCormick, and geometric comparison

For the trilinear monomial $e$6 over a nonnegative box $e$7, double McCormick applies the standard McCormick inequalities twice: first to one bilinear factor, then to the product of the resulting auxiliary variable with the remaining factor. Grouping $e$8 first and multiplying by $e$9 second yields one of three possible relaxations, denoted $y_{J_e}$0 according to which variable is not grouped first. Speakman and Lee derive closed-form four-dimensional volume formulae for all three relaxations and for the exact trilinear hull $y_{J_e}$1 (Speakman et al., 2015).

Under the ordering

$y_{J_e}$2

one has

$y_{J_e}$3

Thus the tightest double-McCormick relaxation is $y_{J_e}$4, while $y_{J_e}$5 is the weakest. The paper emphasizes that these results apply not only to original variables but also to products involving auxiliary variables in a factorable formulation, so they extend directly to recursive convexification choices in spatial branch-and-bound. It also states that the “best” double-McCormick often has volume very close to the full trilinear hull $y_{J_e}$6, while using only $y_{J_e}$7 auxiliary variable and $y_{J_e}$8 inequalities instead of the $y_{J_e}$9 needed for $\{0,1\}^n$00 (Speakman et al., 2015).

Experimental work on box cubic problems substantiates the practical relevance of the volume criterion. For boxcup instances on dense, sparse, and very sparse 3-uniform hypergraphs, the authors solve LP relaxations over $\{0,1\}^n$01 in many random objective directions and compare the resulting quasi mean widths. Across all three sparsity regimes, the ordering

$\{0,1\}^n$02

held almost without exception. Scatter-plot regressions of aggregated idealized radius against $\{0,1\}^n$03 gave $\{0,1\}^n$04-values above $\{0,1\}^n$05, and the worst-case experiments with $\{0,1\}^n$06 confirmed that the maximal gap between the best and bad double-McCormick relaxations occurs when $\{0,1\}^n$07 (Speakman et al., 2016).

This geometric analysis places a quantitative interpretation on recursive McCormick choices: the order of aggregation is not merely representational, but measurably alters the relaxation.

5. Illustrative examples and computational evidence

A canonical example in binary polynomial optimization is

$\{0,1\}^n$08

A simple recursive McCormick sequence factors each cubic as $\{0,1\}^n$09 and introduces three distinct artificial variables $\{0,1\}^n$10, yielding LP-bound $\{0,1\}^n$11. A stronger sequence reuses $\{0,1\}^n$12 through $\{0,1\}^n$13 and $\{0,1\}^n$14; this overlapping recursive McCormick relaxation achieves the sharp bound $\{0,1\}^n$15. The flower relaxation $\{0,1\}^n$16, and therefore also $\{0,1\}^n$17, cuts the same fractional extreme point and gives $\{0,1\}^n$18 (Khajavirad, 2022).

Formulation choice	Structural feature	Reported bound
Sequence 1	three distinct artificial variables $\{0,1\}^n$19	LP-bound $\{0,1\}^n$20
Sequence 2	reuse of $\{0,1\}^n$21 in two products	sharp bound $\{0,1\}^n$22
$\{0,1\}^n$23 / $\{0,1\}^n$24	cuts the same fractional extreme point as Sequence 2	$\{0,1\}^n$25

The example isolates the role of overlap. It explains why recursive McCormick constructions that share intermediate factors can be much stronger than decompositions that treat each monomial independently.

For unconstrained multilinear benchmarks, computational findings for Recursive McCormick Linearization show a structured trade-off among formulation size, bound quality, and overall solution time. On mult3/mult4, vision, and autocorr instances, MinLin and Greedy often reduce the number of artificial variables by $\{0,1\}^n$26–$\{0,1\}^n$27 compared to full sequential (Seq), with MinLin strictly superior to Greedy in linearization size. At the root node, the Best-Bound model (BB) yields the tightest relaxation, followed by MinLin and Greedy, and Seq last. On mult3/mult4, all RMLs close the gap but BB and even the full-model (Full) are fastest. On vision and autocorr, only Full and BB often make nontrivial progress; MinLin and Seq lag behind Greedy on autocorr but outperform it on vision. The exact MinLin formulation solves minimum-size RML to optimality in $\{0,1\}^n$28 on most instances under a $\{0,1\}^n$29 limit, with mult4 instances hardest (Raghunathan et al., 2022).

These benchmark results reinforce a basic methodological point: recursive McCormick should be understood not as a single relaxation, but as a design space of relaxations with different structural objectives.

6. Complexity, implementation, and applied variants

Although the extended flower relaxation contains exponentially many potential cuts, its separation problem can be solved in strongly polynomial time when the rank $\{0,1\}^n$30 of the hypergraph is fixed. The stated procedure enumerates, for each center $\{0,1\}^n$31, all subsets $\{0,1\}^n$32 of its neighbors of size $\{0,1\}^n$33, checks the $\{0,1\}^n$34-condition, and selects the most violated inequality in time $\{0,1\}^n$35, i.e. $\{0,1\}^n$36 for constant $\{0,1\}^n$37. By the equivalence of separation and optimization, $\{0,1\}^n$38 can therefore be optimized in strongly polynomial time for fixed-degree binary polynomials. One stated consequence is that, instead of solving a hard MIP to choose an “optimal” recursive sequence, one can solve or separate the extended flower relaxation and obtain at least as strong a bound (Khajavirad, 2022).

The generalized recursive-linearization framework clarifies the implementation alternatives. $\{0,1\}^n$39 has exponentially many inequalities in general, and separation is NP-hard when degree is part of the input, whereas each recursive McCormick model uses only polynomially many variables and constraints for fixed degree. Two algorithmic viewpoints are therefore available: dynamically generate violated flower inequalities, or dynamically add new recursive linearizations with auxiliary variables and McCormick constraints. Theorem 5.1 shows that these two viewpoints are equivalent in strength (Schutte et al., 2023).

In a different application setting, the 2026 power-systems paper uses the phrase “iterative (or recursive) McCormick relaxation” for a repeated bound-tightening scheme on a bilinear term $\{0,1\}^n$40. Over bounds $\{0,1\}^n$41 and $\{0,1\}^n$42, the standard McCormick envelope is defined by four linear inequalities, and the paper tightens the relaxation by splitting the allowable range of one variable into increments $\{0,1\}^n$43, solving a linearized master problem at each slice, and “re-centering” the bounds after each solve. In the network-impedance application, $\{0,1\}^n$44 is the change in susceptance $\{0,1\}^n$45 and $\{0,1\}^n$46 is the voltage-angle difference $\{0,1\}^n$47; after iteration $\{0,1\}^n$48 the next bounds are

$\{0,1\}^n$49

The paper states that the maximum relaxation error for one McCormick envelope is

$\{0,1\}^n$50

and gives a high-level argument that the aggregate error tends to $\{0,1\}^n$51 as $\{0,1\}^n$52, while also noting that no formal proof is in the paper (Park et al., 20 Feb 2026).

The reported numerical evidence shows the intended effect. For Case 300 with $\{0,1\}^n$53, the original McCormick relaxation has error $\{0,1\}^n$54 and the iterative McCormick method has error $\{0,1\}^n$55; at $\{0,1\}^n$56, the corresponding errors are $\{0,1\}^n$57 and $\{0,1\}^n$58. Larger systems show the same qualitative pattern: Case 588_sdet reports approximately $\{0,1\}^n$59 versus $\{0,1\}^n$60, Case 1354pegase approximately $\{0,1\}^n$61 versus $\{0,1\}^n$62, and Case 1888rte approximately $\{0,1\}^n$63 versus $\{0,1\}^n$64. The paper summarizes this as the recursive McCormick scheme driving the relaxation gap downward by roughly an order of magnitude, or more, with only a modest increase in total solve-time (Park et al., 20 Feb 2026).

Taken together, these lines of work give recursive McCormick a dual status. In binary multilinear optimization, it is a precisely analyzable family of hypergraph-based linearizations whose aggregate power is exactly captured by extended-flower inequalities. In factorable and applied nonconvex optimization, it is also a practical template for recursively or iteratively tightening bilinear relaxations through decomposition, reuse, and local re-centering.