Reformulation Linearization Technique (RLT)

• RLT is a methodology that constructs strong linear or conic relaxations for nonconvex polynomial, quadratic, bilinear, and mixed-integer problems by lifting them into an extended space.
• It generates valid nonlinear constraints by multiplying original constraints with nonnegative factors and linearizing resulting products via auxiliary variables and inequalities.
• RLT operates hierarchically, offering a trade-off between relaxation tightness and model complexity, and finds applications in combinatorial optimization, global polynomial programming, and neural-network verification.

Searching arXiv for recent and foundational RLT-related papers to ground the article. The Reformulation–Linearization Technique (RLT) is a systematic lift-and-linearize methodology for constructing strong linear or conic relaxations of nonconvex polynomial, quadratic, bilinear, and mixed-integer optimization problems. In its classical form, RLT generates new valid nonlinear constraints by multiplying existing constraints and bound factors, introduces auxiliary variables for monomials or products, and then replaces those products by linear or conic relations in an extended space. Across contemporary work, this paradigm appears as a hierarchy of relaxations for $0$–$1$ and mixed-integer sets, as a polyhedral approach to polynomial optimization, as a strengthening device for semidefinite and second-order cone relaxations of quadratically constrained programs, and as a mechanism for constructing sharp set representations and convex hulls in hybrid zonotope models (Gonçalves et al., 2015, Glunt et al., 21 Mar 2025, Jiang et al., 2016, Qiu et al., 2023).

1. Historical lineage and canonical problem classes

Classical RLT was introduced by Adams and Sherali and further developed by Sherali and co-authors as a general methodology for generating strong linear relaxations for $0$–$1$ mixed-integer programs with polynomial, especially quadratic, objective functions or constraints (Gonçalves et al., 2015). In the Sherali–Adams formulation used in recent set-based work, the prototypical mixed-integer set is

$$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$

and RLT constructs a hierarchy $X_d$, $d=1,\dots,n$, by introducing auxiliary variables $w_J=\prod_{j\in J}x_j$ and $v_{J,k}=y_k\prod_{j\in J}x_j$, together with linearized constraints derived from products of the original equations and bound-factor terms (Glunt et al., 21 Mar 2025).

For continuous polynomial optimization, RLT is presented as a polyhedral approach to globally solve nonconvex polynomial optimization problems by lifting them into a higher-dimensional space and replacing nonlinear monomials with linear constraints (Barros-González et al., 19 Jun 2026). A generic continuous polynomial program is written as

$$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$

with $1$0 a hyperrectangle and each $1$1 polynomial (Barros-González et al., 19 Jun 2026). For bounded polynomial programs, a concrete realization is the binary reformulation-and-linearization scheme of “Bounded Global Optimization for Polynomial Programming using Binary Reformulation and Linearization” (Norman, 2012), which transforms the original problem into a pair of mixed binary-linear programs yielding lower and upper bounds on the global optimum.

For quadratic programming, RLT relaxations also arise in box-constrained and linearly constrained nonconvex quadratic programs. In the box-constrained setting, the lifted feasible set is

$1$2

and the RLT relaxation replaces $1$3 by McCormick inequalities, producing a linear program (Qiu et al., 2023). In the more general linearly constrained setting,

$1$4

the RLT relaxation uses pairwise products of inequalities and variable-times-equality products to obtain a lifted polyhedron $1$5 (Qiu et al., 2023).

A separate but closely related line treats RLT as a geometric strengthening tool for set representations. In “Sharp Hybrid Zonotopes: Set Operations and the Reformulation-linearization Technique” (Glunt et al., 21 Mar 2025), RLT is used as a lift-and-project procedure on the factor space of a 01-hybrid zonotope, producing a sharp representation whose convex relaxation equals the convex hull.

2. Core mechanism: reformulation, lifting, and linearization

RLT has two canonical steps. The first is reformulation: multiply existing valid constraints by nonnegative factors such as variables, complements, or bound slacks. The second is linearization: introduce new variables for the resulting products and replace nonlinear expressions by linear or conic constraints in an extended space (Bestuzheva et al., 2022, Gonçalves et al., 2015).

For linearly constrained quadratic programs, if

$1$6

then

$1$7

is valid on the original feasible set. After lifting $1$8 to $1$9, this becomes the linear RLT inequality

$0$0

which tightens the base semidefinite relaxation (Jiang et al., 2016). In box-constrained bilinear or quadratic settings, the same principle yields the classical McCormick envelope. For $0$1, RLT replaces $0$2 by

$0$3

or, equivalently, by the matrix form

$0$4

together with $0$5 (Qiu et al., 2023).

For bounded polynomial programming, the reformulation can proceed through binary expansion of each bounded variable. The paper (Norman, 2012) writes

$0$6

with binary variables $0$7, remainder $0$8, and an upper-bound constraint enforcing $0$9. Monomials are then expanded into products of unit variables and remainders, and auxiliary variables $1$0 are introduced for products such as

$1$1

The paper imposes RLT-like inequalities

$1$2

or, in the pure binary case,

$1$3

to enforce product behavior in a mixed-binary linear model (Norman, 2012).

For mixed-binary nonlinear programs with explicit or implicit bilinear relations, RLT cuts are generated by multiplying linear rows $1$4 with bound factors

$1$5

and linearizing the resulting terms with explicit product variables, binary square identities, clique implications, or McCormick envelopes (Bestuzheva et al., 2022). This extends RLT to settings where bilinear products are encoded only implicitly in a MILP through big-$1$6-type linear constraints (Bestuzheva et al., 2022).

In 01-hybrid zonotope factor spaces, RLT uses the Sherali–Adams variables

$1$7

and linearized expressions

$1$8

producing a lifted set $1$9 whose projection $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$0 tightens the convex relaxation and becomes exact at level $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$1 (Glunt et al., 21 Mar 2025).

3. Hierarchies, exactness, and convergence

RLT is naturally hierarchical. For $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$2 binary variables, the Sherali–Adams construction gives levels $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$3, with

$$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$4

and exact convex hull recovery at the top level: $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$5 (Glunt et al., 21 Mar 2025). In combinatorial problems such as QAP, the same principle appears as RLT1, RLT2, and RLT3, where higher levels multiply constraints by more factors, introduce higher-order product variables, and give tighter LP bounds at sharply higher dimensional cost (Gonçalves et al., 2015).

For the QAP, the level-2 formulation introduces variables $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$6 for pairwise products $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$7 and variables $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$8 for triple products $$X = \{(x,y)\mid Ax + By = \beta,\; x\in\{0,1\}^n,\; y\in[0,1]^m\},$$9, together with complementarity equalities such as

$X_d$0

and

$X_d$1

(Gonçalves et al., 2015). The paper emphasizes the trade-off: RLT3 gives tighter bounds than RLT2, but RLT3 required about $X_d$2 GB of main memory for a QAP instance with $X_d$3, whereas a GPU-based RLT2 scheme achieved up to $X_d$4 times faster runtime and $X_d$5 less memory than the level-3 version on some instances (Gonçalves et al., 2015).

In the generalized moment problem over the simplex, the paper (Kirschner et al., 2021) studies an LP, RLT-type hierarchy

$X_d$6

based on truncated moment functionals $X_d$7, monomial nonnegativity, and the simplex identity

$X_d$8

Using a quantitative version of Pólya’s Positivstellensatz, it proves

$X_d$9

hence a convergence rate of $d=1,\dots,n$0 (Kirschner et al., 2021). The same paper contrasts this with a Lasserre-type SDP hierarchy over the sphere, where the rate is $d=1,\dots,n$1 (Kirschner et al., 2021). This suggests a systematic LP-versus-SDP trade-off: cheaper linear relaxations with slower asymptotic convergence, versus stronger semidefinite relaxations with faster asymptotic convergence.

For box-constrained quadratic programs, exactness of RLT and SDP-RLT can be characterized pointwise by the principal submatrix $d=1,\dots,n$2 associated with the fractional index set $d=1,\dots,n$3. The paper (Qiu et al., 2023) proves: $d=1,\dots,n$4 and

$d=1,\dots,n$5

At the whole-relaxation level, exact RLT is equivalent to the existence of a vertex $d=1,\dots,n$6 such that $d=1,\dots,n$7 is optimal for the relaxation, while exact SDP-RLT is characterized by the existence of a rank-one optimal lifted solution $d=1,\dots,n$8 satisfying the primal-dual optimality system (Qiu et al., 2023).

For general linearly constrained quadratic programs, exactness also has a geometric form. The paper (Qiu et al., 2023) shows that RLT is exact if and only if there exists a point $d=1,\dots,n$9 that lies on a minimal face of $w_J=\prod_{j\in J}x_j$0 such that $w_J=\prod_{j\in J}x_j$1 is an optimal solution of the RLT relaxation. Moreover, every point on the same minimal face is then optimal for the original QP (Qiu et al., 2023). The same paper links recession directions, boundedness, and vertices of the original polyhedron and the RLT polyhedron, proving, for example, that $w_J=\prod_{j\in J}x_j$2 is nonempty and bounded if and only if the RLT feasible region $w_J=\prod_{j\in J}x_j$3 is nonempty and bounded (Qiu et al., 2023).

A more delicate convergence issue appears in continuous polynomial optimization. “A note on the convergence guarantees of RLT-based algorithms for polynomial optimization” (Barros-González et al., 19 Jun 2026) identifies a mathematical oversight in a foundational redundancy result for bound-factor constraints: deducing

$w_J=\prod_{j\in J}x_j$4

from

$w_J=\prod_{j\in J}x_j$5

requires $w_J=\prod_{j\in J}x_j$6, not merely $w_J=\prod_{j\in J}x_j$7. The paper gives a counterexample where a zero-length variable interval causes degree-$w_J=\prod_{j\in J}x_j$8 bound-factor constraints to fail to imply lower-degree ones, producing an unbounded child-node relaxation even though the original bounded problem remains well posed (Barros-González et al., 19 Jun 2026). It then shows that correctness is recovered by imposing the natural assumption $w_J=\prod_{j\in J}x_j$9 for all active variables, or by structurally eliminating fixed variables and regenerating the reduced RLT formulation (Barros-González et al., 19 Jun 2026).

4. Geometry, convex hulls, and relations to competing relaxations

RLT has a precise geometric interpretation as a closure operation in extended space. For binary mixed-integer sets $v_{J,k}=y_k\prod_{j\in J}x_j$0, the weak and strong first-level RLT closures are

$v_{J,k}=y_k\prod_{j\in J}x_j$1

and satisfy

$v_{J,k}=y_k\prod_{j\in J}x_j$2

(Hof et al., 17 Nov 2025). In the pure $v_{J,k}=y_k\prod_{j\in J}x_j$3–$v_{J,k}=y_k\prod_{j\in J}x_j$4 case $v_{J,k}=y_k\prod_{j\in J}x_j$5, the strong and weak closures coincide and depend only on the intersection of $v_{J,k}=y_k\prod_{j\in J}x_j$6 with the boundary of the cube $v_{J,k}=y_k\prod_{j\in J}x_j$7 (Hof et al., 17 Nov 2025).

A central geometric characterization states that, for pure $v_{J,k}=y_k\prod_{j\in J}x_j$8–$v_{J,k}=y_k\prod_{j\in J}x_j$9 problems, $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$0 belongs to the RLT closure if and only if for each fractional coordinate $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$1 there exist points $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$2 with $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$3 such that

$$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$4

together with compatibility equations coupling these decompositions across different fractional variables (Hof et al., 17 Nov 2025). This clarifies why RLT dominates single-variable lift-and-project disjunctions: RLT simultaneously enforces consistency among all such local decompositions.

Indeed, the same paper proves

$$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$5

where $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$6 is the lift-and-project closure based on disjunctions $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$7 for $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$8 (Hof et al., 17 Nov 2025). More strongly, if $$\begin{aligned} \text{minimize}\quad & \phi_0(x)\ \text{subject to}\quad & \phi_r(x)\ge \beta_r,\quad r=1,\dots,m_1,\ & \phi_r(x)=\beta_r,\quad r=m_1+1,\dots,m,\ & x\in\Omega\subset\mathbb{R}^{|N|}, \end{aligned}$$9 satisfies a valid cardinality equation

$1$00

then the strong RLT closure dominates the corresponding disjunctive hull

$1$01

(Hof et al., 17 Nov 2025). By contrast, RLT does not dominate disjunctive programming based on cardinality inequalities

$1$02

the paper gives a counterexample where a point lies in $1$03 but violates the associated disjunctive hull (Hof et al., 17 Nov 2025).

For bilinear convexification, RLT coincides with the convex hull when only box bounds are present. For

$1$04

the convex hull $1$05 is given by the four RLT constraints

$1$06

When explicit bounds $1$07 are added, the convex hull remains compactly representable: with only an upper bound on $1$08, RLT plus $1$09 and the SOC

$1$10

suffices; with only a lower bound on $1$11, RLT plus $1$12 and one SOC suffices; with both bounds, no more than three SOC constraints are needed, each applicable on a subset of $1$13-space (Anstreicher et al., 2020). This stands in contrast to earlier infinite families of lifted tangent inequalities, and it shows that RLT can serve as the polyhedral core of an exact SOC-representable hull description (Anstreicher et al., 2020).

RLT also interacts nontrivially with semidefinite and second-order cone relaxations. In AC optimal power flow, a conic relaxation called TCR is built by combining semidefinite optimization with RLT. The paper (Bingane et al., 2019) uses the slack-bus RLT inequality

$1$14

together with local $1$15 PSD constraints

$1$16

obtaining a relaxation stronger than SOCR and dominated by SDR (Bingane et al., 2019). On large test cases, TCR was significantly tighter than SOCR and much faster than chordal SDP, with average cost-minimization gaps $1$17 for TCR versus $1$18 for SOCR and $1$19 for CHR/SDR on large-scale instances (Bingane et al., 2019).

In nonconvex QCQP, SOC-RLT and generalized SOC-RLT extend the RLT idea to second-order cone and semidefinite structures. For linear constraints $1$20, classical RLT adds

$1$21

For convex quadratic constraints written as SOCs, SOC-RLT multiplies an SOC by a nonnegative linear factor and linearizes the product; for nonconvex quadratics decomposed as differences of PSD forms $1$22, the paper (Jiang et al., 2016) introduces GSRT-A and GSRT-B constraints that use auxiliary variables $1$23, SOC representations, and linearized squared equalities. It proves the bound chain

$1$24

and reports that GSRT-B is typically tighter and faster than GSRT-A when the range condition $1$25 holds (Jiang et al., 2016).

In QCQP with multiple ball constraints,

$1$26

RLT appears as the Kronecker-product inequalities $1$27 and the spectral-norm-based inequalities $1$28. The paper (Kılınç-Karzan et al., 2024) proves that Burer’s lifted cone $1$29, a compact relaxation interpretable as a particular case of the moment-sum-of-squares hierarchy, implies both the Kronecker RLT inequalities and the Zhen et al. inequalities. In other words,

$1$30

so in this structured ball-constrained setting the RLT inequalities are strictly dominated by a stronger lifted moment-SOS relaxation (Kılınç-Karzan et al., 2024). This does not negate RLT’s utility; rather, it locates RLT as a first-order or structured subset of a richer convexification framework (Kılınç-Karzan et al., 2024).

5. Applications across optimization, verification, and combinatorial structure

RLT has been applied to a wide range of optimization and verification tasks. In bounded polynomial optimization, the binary-expansion method of (Norman, 2012) constructs optimistic and pessimistic mixed binary linear programs,

$1$31

whose solutions provide lower and upper bounds on the global optimum. The method quantifies linearization errors through explicit bounds

$1$32

and tightens those bounds as the user-selected tolerances $1$33 decrease (Norman, 2012). The paper reports, for example, that a mixed discrete/continuous engineering design problem is solved exactly with interval $1$34, and that a Sherali–Tuncbilek test problem is certified to have global minimum $1$35 after interval refinement (Norman, 2012).

In mixed-integer nonlinear programming, efficient separation of RLT cuts has become a major algorithmic topic. “Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Terms” (Bestuzheva et al., 2022) develops a theorem for detecting implicit bilinear products from pairs of linear constraints involving a binary variable, and a row-marking separation algorithm that preselects promising row–factor combinations based on current LP violation patterns. Computationally, row marking reduced the fraction of total solve time spent in RLT separation from about $1$36 to about $1$37 on MILPs and from about $1$38–$1$39 to about $1$40–$1$41 on MINLPs, while retaining the strength of the cuts on hard instances (Bestuzheva et al., 2022).

In set-based reachability and neural-network verification, RLT is used to produce sharp hybrid zonotopes. Starting from a 01-hybrid zonotope $1$42, the paper (Glunt et al., 21 Mar 2025) applies Sherali–Adams RLT to the factor space and then reinterprets the resulting extended set as a new hybrid zonotope. At full level $1$43, the resulting representation is sharp, meaning its convex relaxation equals $1$44 (Glunt et al., 21 Mar 2025). A numerical example constructs the unsafe level set

$1$45

of a feedforward ReLU network, reduces its initial hybrid zonotope complexity from $1$46 to $1$47, and then applies full RLT to obtain a sharp representation with complexity $1$48, whose convex relaxation exactly equals the convex hull of the unsafe region (Glunt et al., 21 Mar 2025). An intermediate level $1$49 gives a much smaller representation $1$50 whose relaxation area is only $1$51 larger than the exact convex hull (Glunt et al., 21 Mar 2025).

In binary quadratic optimization, the linearization problem provides a different lens on RLT. The paper (Hu et al., 2018) defines a matrix $1$52 to be linearizable if there exists $1$53 with

$1$54

for a binary feasible set $1$55. It proves that the first-level RLT bound $1$56 is a linearization-based bound and shows the hierarchy

$1$57

when $1$58 is redundant (Hu et al., 2018). For the quadratic shortest path problem on directed acyclic graphs, it gives a polynomial-time algorithm for the linearization problem and a full characterization of the space of linearizable matrices, enabling the strongest linearization-based bound $1$59 (Hu et al., 2018).

In the QAP, RLT remains one of the strongest LP-based bounding methodologies. The GPU-based RLT2 dual ascent method of (Gonçalves et al., 2015) solved the instances tai35b and tai40b exactly for the first time, producing exact optimal values $1$60 and $1$61, respectively (Gonçalves et al., 2015). The same paper reports root-node gaps of $1$62 for tai35b and $1$63 for tai40b, substantially tighter than the $1$64 and $1$65 gaps previously listed in QAPLIB (Gonçalves et al., 2015).

6. Computational structure, scalability, and current directions

The main computational characteristic of RLT is a sharp trade-off between bound strength and model size. In bounded polynomial programming with binary reformulation, the number of unit-product variables and linking constraints grows polynomially in the degree but exponentially in the discretization parameter $1$66; the paper (Norman, 2012) gives rough bounds

$1$67

and notes that branch-and-bound complexity is exponential in the number of binary variables (Norman, 2012). In hybrid zonotopes, full RLT at level $1$68 yields complexity

$1$69

illustrating the same exponential blow-up in the number of binary factors (Glunt et al., 21 Mar 2025). In QAP, RLT2 involves $1$70 variables and coefficients, whereas RLT3 requires an $1$71-matrix of dimension

$1$72

making it memory-intensive even for moderate $1$73 (Gonçalves et al., 2015).

This scalability pressure explains several modern RLT trends. One trend is selective cut generation rather than explicit generation of all RLT constraints. The row-marking and projection-filtering methods of (Bestuzheva et al., 2022) fit this pattern. Another trend is hybridization with stronger convex relaxations. In AC-OPF, TCR keeps only a slack-bus RLT inequality and localized $1$74 PSD blocks, attaining near-SDP tightness at far lower runtime than full chordal SDP (Bingane et al., 2019). In ball-constrained QCQP, Burer’s compact moment-SOS lift strictly subsumes known Kronecker and Zhen-type RLT constraints (Kılınç-Karzan et al., 2024). This suggests that in highly structured quadratic settings, tailored compact semidefinite or moment relaxations can dominate RLT, while in broader combinatorial or mixed-bilinear settings RLT remains a practical and often dominant strengthening device.

A further direction is geometric comparison against disjunctive methods. The domination results of (Hof et al., 17 Nov 2025) show that RLT is stronger than lift-and-project based on single-variable disjunctions and even dominates DP based on cardinality equations with right-hand side $1$75, while failing to dominate DP based on cardinality inequalities with right-hand side $1$76. This suggests that the effective geometry of first-level RLT is closely tied to one-hot or assignment-type structures rather than general at-most-one disjunctions (Hof et al., 17 Nov 2025). A plausible implication is that problems built from assignment, partitioning, or exact-one constraints, such as QAP and some neural-network activation encodings, are especially natural domains for RLT-based strengthening.

Recent theoretical work also indicates that convergence guarantees must be interpreted carefully. The correction in (Barros-González et al., 19 Jun 2026) shows that dropping lower-degree bound-factor constraints is safe only when active variable intervals remain strictly positive or fixed variables are eliminated. This nuance matters particularly in mixed-integer branch-and-bound nodes where variables are routinely fixed to binary values (Barros-González et al., 19 Jun 2026).

Overall, RLT is best understood as a family of lifted polyhedral or conic closures rather than a single formulation. In some settings it gives exact convex hulls; in others it serves as a strong but incomplete first-order approximation; and in still others it functions as a structural component inside SDP, SOC, or hybrid-set formulations. Across these regimes, its central mechanism remains unchanged: multiply valid constraints by nonnegative factors, linearize the resulting products in an extended space, and exploit the induced geometry to tighten global optimization relaxations (Norman, 2012, Glunt et al., 21 Mar 2025, Bestuzheva et al., 2022, Hof et al., 17 Nov 2025).