McCormick Relaxation in Optimization

• McCormick Relaxation is a convexification technique that constructs the tightest convex and concave envelopes for bilinear and multilinear terms over bounded domains.
• It is implemented via recursive linearization, where auxiliary variables and careful grouping create lifted polyhedral relaxations that enhance solver performance.
• Practical applications in network flow, energy systems, and robotics benefit from improvements like bound-tightening and structural enhancements to further close the relaxation gaps.

McCormick Relaxation is a fundamental technique for convexifying nonconvex polynomial and bilinear terms in mathematical optimization, underpinning many high-performance solvers for nonlinear, mixed-integer, and multilinear programs. At its core, the McCormick relaxation constructs the tightest possible convex and concave envelopes for products of variables over box or polyhedral domains, which can be efficiently translated into linear or convex quadratic constraints. This approach enables tractable relaxations of inherently hard optimization problems, yielding globally-valid bounds that facilitate branch-and-bound, cutting-plane, and decomposition algorithms across domains such as network flow, energy systems, robotics, and PDE-constrained optimization.

1. Classical McCormick Envelope and Basic Properties

For a bilinear term $z = x y$ with $x \in [x^L, x^U]$, $y \in [y^L, y^U]$, the convex hull of the graph is given by four linear inequalities:

$$\begin{aligned} z &\geq x^L y + y^L x - x^L y^L \ z &\geq x^U y + y^U x - x^U y^U \ z &\leq x^U y + y^L x - x^U y^L \ z &\leq x^L y + y^U x - x^L y^U \end{aligned}$$

These inequalities precisely define the polyhedral relaxation for $z = x y$ on the rectangle $[x^L, x^U] \times [y^L, y^U}$ (Boland et al., 2015, Speakman et al., 2015, Khademnia et al., 2023). In the $[0,1]$ binary case, they reduce to $z \geq 0$, $z \geq x + y - 1$, $z \leq x$, $z \leq y$. The McCormick envelope is provably the tightest convex relaxation for a single bilinear or multilinear term over such domains.

2. Recursive McCormick Linearization and Multilinear Programs

For multilinear monomials (e.g., $x_1 x_2 x_3$), the McCormick relaxation is applied recursively via the introduction of auxiliary variables representing intermediate products, yielding a lifted polyhedral relaxation. There are multiple choices for how to group variables; each sequence of groupings (“recursive McCormick linearizations,” RMLs) produces a distinct relaxation in the extended $(x, y)$ variable space. The strength and computational size of the relaxation depend strongly on this recursive sequence (Raghunathan et al., 2022, Khajavirad, 2022).

Algorithmic construction proceeds by decomposing each multilinear term into its bilinear sub-products, linearizing via McCormick envelopes, and enforcing consistency with the original variables. The problem of finding the smallest RML or one with the best relaxation bound is combinatorially hard (NP-hard for degree 3), but special cases are tractable via kernelization and fixed-parameter approaches.

Table: Properties of RML Strategies (Raghunathan et al., 2022)

Strategy	Auxiliary Variable Count	LP Bound Strength
Seq	High	Variable
Greedy	Moderate	Improved over Seq
MinLin	Lowest	Strongest LP bounds
Full	Highest	Maximum strength

For polynomial programs with binary variables, all recursive McCormick relaxations are dominated in bound quality by the so-called extended flower relaxation (Schutte et al., 2023, Khajavirad, 2022).

3. Volume-Based Comparison and Double-McCormick for Trilinear Terms

When convexifying trilinear monomials (e.g., $x_1 x_2 x_3$), applying McCormick envelopes twice leads to three possible “double-McCormick” relaxations, corresponding to different groupings. The strength of each can be quantified via their associated polytope volume; the relaxation with the lowest volume provides the tightest possible bound aside from the convex hull (Speakman et al., 2015, Speakman et al., 2016). Closed-form formulae are given for the volumes of each grouping and of the true hull; in nearly all settings, grouping the pair with the tightest box product first produces the strongest relaxation.

4. Strength, Gaps, and Tightness of McCormick Relaxations

The relaxation gap—the difference between the McCormick relaxation and the true convex hull—is a central metric. For random bilinear forms over $n$ variables, Boland et al. proved that the worst-case gap scales as $\Theta(\sqrt{n})$ and cannot be bounded by any constant independent of problem size (Boland et al., 2015). The exactness of the McCormick relaxation is characterized: cycles in the underlying problem graph containing odd sign combinations of coefficients yield relaxation gaps, while graphs that are bipartite (for all-positive coefficients) or whose cycles have all even sign pairs produce exact hull relaxations.

In major applications (e.g., gas network flows, energy planning, mixed-integer PDEs), empirical results show that with carefully chosen variable bounds, iterative bound-tightening, or domain-specific structure exploitation, McCormick relaxations can be made extremely tight—even reaching exactness under appropriate conditions (Singh et al., 2019, Deng et al., 2020, Zhou et al., 2023, Leyffer et al., 12 Jun 2024).

5. Bound-Tightening, Strengthening, and Structural Enhancements

Several techniques strengthen McCormick relaxations:

• Subgradient-based tightening: Propagate affine under- and overestimators along the computational graph, minimize/maximize them over variable boxes, and update range bounds for each factor, yielding systematically improved relaxations with very low overhead. This approach is guaranteed to never worsen bounds and is effective on small or medium-scale problems (Najman et al., 2017).
• Bound-tightening algorithms: Iterative methods (e.g., STBA, OBBT) solve relaxed programs, successively shrink variable bounds, and update McCormick envelopes until the relaxation error converges below user tolerance (Zhou et al., 2023, Deng et al., 2020, Leyffer et al., 12 Jun 2024).
• Partitioned piecewise McCormick: Domain partitioning focused around regions of interest, possibly via dynamic iterative refinement, yields sparse lifted relaxations with dramatically fewer binary variables and much tighter bounds (Nagarajan et al., 2016).

Strengthening via lifted valid inequalities, network-structure cuts, or extended flower inequalities further closes relaxation gaps, sometimes matching the true convex hull in network-constrained or combinatorial settings (Khademnia et al., 2023, Schutte et al., 2023, Khajavirad, 2022, Pia et al., 17 Jul 2025).

6. Practical Applications and Empirical Performance

McCormick relaxations have penetrated multiple application domains:

• Gas flow and expansion planning: Both steady-state gas flow and expansion MINLPs are convexified via exact McCormick envelopes for binary-continuous bilinear products, with theoretical and empirical near-exactness over thousands of test cases (Singh et al., 2019, Borraz-Sanchez et al., 2015).
• Energy networks and district heating: Reformulations isolate the core bilinear term, which is convexified by McCormick relaxations and improved via iterative bound tightening, enabling fast and globally valid planning on large real-world systems (Deng et al., 2020).
• Robot motion planning: Data-driven envelope learning combined with two-stage convex decomposition delivers sub-second plan times with guaranteed feasibility for high-DOF, complex terrain navigation (Lin et al., 2021).
• OPF and DC networks: McCormick relaxations, especially when combined with advanced bound-tightening, yield convex and exact global solutions in DC power distribution (Zhou et al., 2023).
• PDE-constrained design: Locally averaged McCormick relaxations reduce infinite constraint sets to finite ones, provide certified error bounds scaling with mesh size, and converge (in the sense of Γ-convergence) to the continuous relaxed limit (Leyffer et al., 12 Jun 2024).
• Binary polynomial and combinatorial optimization: McCormick relaxations, while extensible via recursive and complete-edge strategies, see their practical bound quality dominated by flower and RLT-type inequalities for complex hypergraphs (Pia et al., 17 Jul 2025).

7. Advanced Topics: Network Polytopes, SDP-comparisons, and Solver Integration

When bilinear terms are embedded in network or assignment polytopes, the classical McCormick relaxation is generally not tight; explicit tree- and forest-based facet inequalities are needed to recover the true convex hull (Khademnia et al., 2023). McCormick relaxations in separable bilinear programs are provably as strong as any SDP enhancement; global solvers initialized with McCormick + lifted bilinear cuts close significant portions of the duality gap at near-zero computational cost (Gu et al., 2022).

Solver integration proceeds via embedding McCormick envelope constraints and bound-tightening loops in branch-and-bound frameworks, LP-based separation routines, or dynamic variable/constraint addition, with substantial improvements in node gap closure, convergence rate, and runtime observed empirically in MINLPLib-scale benchmarks (Müller et al., 2019, Nagarajan et al., 2016).

8. Structural Limitations, Extensions, and Theoretical Landscape

While the McCormick relaxation is optimal for a single or small number of uncoupled bilinear products, its strength deteriorates for larger, interconnected multilinear or network-linked programs unless fortified by advanced cuts or extended relaxations. The flower and complete edge relaxations, which generalize and dominate recursive McCormick relaxations, offer polynomial-size models for fixed degree and capture all known facet-defining inequalities for binary multilinear polytopes (Schutte et al., 2023, Khajavirad, 2022, Pia et al., 17 Jul 2025). Dynamic strategies such as row generation and addition of extended variables balance relaxation size and separation tractability; hypergraph acyclicity governs conditions for exactness.

An important direction for future research remains the analytic and computational extension of these envelope techniques to higher-degree monomials, domain-adapted partitioning, and the integration with hybrid global-local algorithms in more complex settings. Empirical evidence corroborates the essential role of McCormick relaxations as both a theoretical and algorithmic primitive throughout large-scale optimization and control.