Relaxation strength for multilinear optimization: McCormick strikes back

TL;DR

This work addresses tightening relaxations for multilinear optimization problems by comparing the extended flower relaxation $\operatorname{FR}(G)$ with relaxations arising from recursive McCormick linearizations. It extends Khajavirad's result by showing a broader equivalence: the strength of $\operatorname{FR}(G)$ can be replicated by intersecting projected relaxations from multiple McCormick linearizations, and FR is projection-stable under subgraph reductions. The authors provide a simpler proof of the dominance result and prove a converse via projection arguments, establishing that $\operatorname{FR}(G)=\bigcap_{\mathcal{D}} P_{\mathcal{E}}(\mathcal{D})$ for all recursive linearizations and, in particular, for all recursive McCormick linearizations. These findings clarify when flower-based and McCormick-based approaches deliver equivalent tightening power, while highlighting practical considerations such as NP-hard flower separation and the potential for running intersection inequalities to strengthen intersections.

Abstract

We consider linear relaxations for multilinear optimization problems. In a recent paper, Khajavirad proved that the extended flower relaxation is at least as strong as the relaxation of any recursive McCormick linearization (Operations Research Letters 51 (2023) 146-152). In this paper we extend the result to more general linearizations, and present a simpler proof. Moreover, we complement Khajavirad's result by showing that the intersection of the relaxations of such linearizations and the extended flower relaxation are equally strong.

Figures (4)

• Figure 1: Three linearizations for the minimization problem in \ref{['fig_linear_problem']} with the depicted hypergraph $G = (V, \mathcal{E})$ with $V = \{1,2,3,4\}$ and $\mathcal{E} = \{ \color{red}{\{1,2,3\}}, \color{green!60!black}{\{2,3,4\}}, \color{purple}{\{1,2\}}, \color{blue}{\{2,3\}} \}$. The arcs that leave a node $I \subseteq V$ indicate the product that is used to represent $z_I$. Consider the point $z^{(1)} \in \mathbb{R}^{\mathcal{S} \cup \mathcal{E}}$ with $z^{(1)}_{\color{green!60!black}{\{2,3,4\}}} = 0$, $z^{(1)}_{\color{red}{\{1,2,3\}}} = z^{(1)}_{\color{purple}{\{1,2\}}} = z^{(1)}_{\{1\}} = z^{(1)}_{\{2\}} = z^{(1)}_{\{3\}} = \tfrac{1}{2}$ and $z^{(1)}_{\{4\}} = 1$. It is contained in $P(\mathcal{D}^{\text{(b)}})$. However, it is contained in neither $P_{\mathcal{E}}(\mathcal{D}^{\text{(c)}})$ nor in $P_{\mathcal{E}}(\mathcal{D}^{\text{(d)}})$ since in both linearizations the arc from $\color{red}{\{1,2,3\}}$ to $\color{blue}{\{2,3\}}$ implies $z^{(1)}_{\color{blue}{\{2,3\}}} \geq \tfrac{1}{2}$ and since $z^{(1)}_{\color{green!60!black}{\{2,3,4\}}} + (1 - z^{(1)}_{\color{blue}{\{2,3\}}}) + (1 - z^{(1)}_{\{4\}}) \geq 1$ implies $z^{(1)}_{\color{blue}{\{2,3\}}} \leq 0$. Consider the point $z^{(2)} \in \mathbb{R}^{\mathcal{E} \cup \mathcal{S}}$ with $z^{(2)}_{\color{red}{\{1,2,3\}}} = 0$, $z^{(2)}_{\color{green!60!black}{\{2,3,4\}}} = z^{(2)}_{\color{purple}{\{1,2\}}} = z^{(2)}_{\{2\}} = z^{(2)}_{\{3\}} = z^{(2)}_{\{4\}} = \tfrac{1}{2}$ and $z^{(2)}_{\{1\}} = 1$. It is contained in $P(\mathcal{D}^{\text{(b)}})$ and in $P_{\mathcal{E}}(\mathcal{D}^{\text{(d)}})$. However, it is not contained in $P_{\mathcal{E}}(\mathcal{D}^{\text{(c)}})$ since the arc from $\color{green!60!black}{\{2,3,4\}}$ to $\color{blue}{\{2,3\}}$ implies $z^{(2)}_{\color{blue}{\{2,3\}}} \geq \tfrac{1}{2}$ and since $z^{(2)}_{\color{red}{\{1,2,3\}}} + (1 - z^{(2)}_{\color{blue}{\{2,3\}}}) + (1 - z^{(2)}_{\{1\}}) \geq 1$ implies $z^{(2)}_{\color{blue}{\{2,3\}}} \leq 0$. Consider the point $z^{(3)} \in \mathbb{R}^{\mathcal{E} \cup \mathcal{S}}$ with $z^{(3)}_{\color{purple}{\{1,2\}}} = 0$ and $z^{(3)}_{\color{red}{\{1,2,3\}}} = z^{(3)}_{\color{green!60!black}{\{2,3,4\}}} = z^{(3)}_{\{1\}} = z^{(3)}_{\{2\}} = z^{(3)}_{\{3\}} = z^{(3)}_{\{4\}} = \tfrac{1}{2}$. It is contained in $P(\mathcal{D}^{\text{(b)}})$ and in $P_{\mathcal{E}}(\mathcal{D}^{\text{(c)}})$. However, it is not contained in $P_{\mathcal{E}}(\mathcal{D}^{\text{(d)}})$ since the arc from $\color{red}{\{1,2,3\}}$ to $\color{purple}{\{1,2\}}$ implies $\tfrac{1}{2} = z^{(3)}_{\color{red}{\{1,2,3\}}} \leq z^{(3)}_{\color{purple}{\{1,2\}}} = 0$.
• Figure 2: An instance in which several extended flower inequalities are captured by one recursive McCormick linearization. The hypergraph $G = (V,\mathcal{E})$ has $V = \{ \{v_i\} \mid i = 1,2,\dotsc,k \} \cup \{ \{u_1, u_2\} \}$ and $\mathcal{E} = \{ \{ u_1, u_2, v_i \} \mid i = 1,2,\dotsc,k \}$ (for $k=12$). It has $k(k-1)$ non-redundant extended flower inequalities by considering each edge as the center edge and choosing any other edge as the unique neighbor. However, it admits a recursive McCormick linearization with one additional variable $z_{\{u_1, u_2\}}$ and $2k+3$ extra inequalities ($z_{\{u_1,u_2,v_i\}} \leq z_{\{u_1,u_2\}}$ and $z_{\{u_1,u_2,v_i\}} + (1-z_{\{u_1,u_2\}}) + (1-z_{v_i}) \geq 1$ for $i=1,2,\dotsc,k$ as well as those for $z_{\{u_1,u_2\}} = z_{u_1} \cdot z_{u_2}$), which turn $3k$ inequalities ($z_{\{u_1,u_2,v_i\}} \leq z_{u_1}$, $z_{\{u_1,u_2,v_i\}} \leq z_{u_2}$ and $z_{\{u_1,u_2,v_i\}} + (1-z_{u_1} + (1-z_{u_2}) + (1-z_{v_i}) \geq 1$ for $i=1,2,\dotsc,k$) from the standard relaxation redundant. By \ref{['thm_equivalent']}, both formulations have the same strength.
• Figure 3: Recursive linearizations showing that being a McCormick linearization is a restriction.
• Figure 4: Hyperedges for a flower inequality with four neighbors and the corresponding incomplete linearization from the proof that \ref{['thm_equivalent_mccormick']} is contained in \ref{['thm_equivalent_flower']}.

Theorems & Definitions (17)

• Definition 1: extended flower inequalities, extended flower relaxation
• Proposition 2
• proof
• Lemma 3
• proof
• Definition 4: recursive linearizations
• Definition 5: relaxation, projected relaxation
• Lemma 6
• proof
• Proposition 7