Geometry of the Reformulation-Linearization-Technique: Domination of Disjunctions

TL;DR

This paper investigates the geometry and strength of the Reformulation-Linearization Technique (RLT) in binary mixed-integer programs, contrasting it with disjunctive programming (DP) and lift-and-project (L&P). It provides a geometric characterization of points in the RLT closure, showing that they can be described as convex combinations on faces of the unit cube subject to an extra constraint, and demonstrates that RLT can dominate DP approaches based on cardinality equations with RHS $1$, with implications for the quadratic assignment problem (QAP). The results extend to a general dominance framework, illustrating when RLT-based relaxations subsume certain disjunctive hulls, and clarify a boundary where domination may fail (e.g., for $\le 1$ cardinality constraints). The work advances understanding of RLT’s strength and paves the way for refined formulations and open geometric questions, with practical impact on QAP formulations and related discrete optimization problems.

Abstract

The reformulation-linearization-technique (RLT) is a well-known strengthening technique for binary mixed-integer optimization. It is well known to dominate lift-and-project strengthening, which is based on disjunctive programming (DP) for single-variable disjunctions. In contrast to the latter, the geometry of RLT is not understood completely. We provide some insights by characterizing the points in the corresponding RLT closure geometrically. We exploit this insight to show that RLT even dominates DP approaches based on cardinality equations with right-hand side 1. This is in contrast to cardinality inequalities with right-hand side 1, whose DPs are not dominated. Our results have applications in the strength comparison for the quadratic assignment problem.

Figures (4)

• Figure 1: Consider the polytope $P \subseteq [0,1]^2$ bounded by the dotted lines and the lines$\bar{Z}^{(1,0)}$ and $\bar{Z}^{(2,0)}$ for $B = \{1,2\}$ for which the points $\bar{z}^{(1,1)},~\bar{z}^{(2,1)}$ and all points $\bar{z}^{(i,0)} \in \bar{Z}^{(i,0)}$ for $i=1,2$ satisfy statement \ref{['enum_rlt_lift_and_project']}. Then \ref{['eq_rlt_lift_and_project']} implies that all $x$ must lie in the shaded region, while \ref{['eq_rlt_extra1_for2']} implies that all $x$ must be on the line through $(0,0)^{\intercal}$ and $(\bar{z}^{(2,1)}_1,\bar{z}^{(1,1)}_2)^{\intercal}$. Consequently, the dashed line segment indicates all points $x \in P$ that satisfy \ref{['enum_rlt_lift_and_project']} for $z^{(i,1)}$ and $z^{(i,0)} \in \bar{Z}^{(i,0)}$ with $i=1,2$.
• Figure 2: Consider the polytope $P \coloneqq \{ x \in [0,1]^2 \mid -2x_1 + x_2 \leq 0,~ 2x_1 + x_2 \leq 2 \}$ for $B = \{1\}$ (depicted as the solid triangle). By construction, the points $\bar{z}^{(1,\beta)}$ from \ref{['thm_rlt_characterization']}\ref{['enum_rlt_lift_and_project']} and \ref{['enum_rlt_equations']} are unique and satisfy $\bar{z}^{(1,0)} = (0,0)^{\intercal}$ and $\bar{z}^{(1,1)} = (1,0)^{\intercal}$. The point $x = (\frac{1}{2},1)^{\intercal}$, for example, together with $\bar{z}$ satisfies statement \ref{['enum_rlt_equations']} but violates statement \ref{['enum_rlt_lift_and_project']}. All points satisfying statement \ref{['enum_rlt_lift_and_project']}, that is $\mathcal{R}_B(P)$, are depicted as the dashed line, while all points in the shaded region satisfy statement \ref{['enum_rlt_equations']}.
• Figure 3: Consider the polytope $P \subseteq \mathbb{R}^N$ with $N = \{1,2,3,4\}$ defined as the convex hull of the three vertices in \ref{['fig_landp_does_not_dominate_first']} and the three vertices in \ref{['fig_landp_does_not_dominate_second']}. On the one hand, the equation $x_1 + x_2 + x_3 = 1$ is valid for $P$, and hence \ref{['thm_cardinality_equation']} shows that $\widehat{\mathcal{R}}_N(P)$ is (contained in and thus) equal to the disjunctive hull for the disjunction $x_1 = 1 \lor x_2 = 1 \lor x_3 = 1$, which is the triangle in \ref{['fig_landp_does_not_dominate_first']}. On the other hand, the point $x^\star = ( \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{2}{3} ) ^{\intercal}$ belongs to $\mathcal{L}_N(P)$.
• Figure 4: We consider $B = N = \{1,2,3,4\}$ and the polytope $P \coloneqq \{ x \in [0,1]^N \mid \textcolor{red}{x_1 + x_2 + x_3 + x_4 \leq 1},~ \textcolor{blue}{2x_1 + 2x_2 + 2x_3 + x_4 \geq 1} \}.$The section of $P$ at $x_1 = 0$ is depicted in \ref{['fig_rlt_does_not_dominate_ineq_picture']}. The associated disjunction for $D = \{1,2,3\}$ is given by $x_1 = 1 \lor x_2 = 1 \lor x_3 = 1 \lor (x_1 = x_2 = x_3 = 0),$ that is, $\bar{X}$ consists of the three unit vectors and the origin in $\mathbb{R}^D$. It is well-known that $\operatorname{conv}(\bar{X})$ is described by nonnegativity constraints and the inequality $x_1 + x_2 + x_3 \leq 1$, which is valid for $P$. Hence, $P \subseteq \operatorname{conv}(\bar{X}) \times [0,1]$. The point $\hat{x}$ specified in \ref{['fig_rlt_does_not_dominate_ineq_point']} lies in $\widehat{\mathcal{R}}_B(P)$ since it can be lifted to $(\hat{x},\hat{y}) \in \widehat{\mathcal{R}} ^{\textup{ext}}_B(P)$. The RLT inequalities \ref{['eq_rlt_first']} and \ref{['eq_rlt_second']} are inequalities \ref{['eq_mccormick']} and those in \ref{['fig_rlt_does_not_dominate_ineq_rlt']}. However, $\hat{x}$ does not lie in the disjunctive hull, which is $\{ x \in [0,1]^4 \mid x_1 + x_2 + x_3 + x_4 = 1 \}$.

Theorems & Definitions (17)

• proposition thmcounterproposition: Balas Balas74
• proposition thmcounterproposition
• proof
• proposition thmcounterproposition
• proof
• theorem thmcountertheorem
• proof
• remark thmcounterremark
• corollary thmcountercorollary
• theorem thmcountertheorem