Piecewise McCormick Relaxations

• Piecewise McCormick relaxations are advanced techniques that partition variable domains to create tighter convex approximations for bilinear and multilinear nonconvexities.
• They employ disjunctive and mixed-integer encodings with dynamic partitioning and projection methods to mitigate exponential scaling and computational overhead.
• These relaxations are pivotal in global optimization frameworks, offering convergence guarantees and significant bound tightening in MINLP, polynomial, and PDE-constrained settings.

Piecewise McCormick relaxations are advanced polyhedral outer approximations for bilinear and multilinear terms in nonconvex optimization, constructed by partitioning underlying variable domains and imposing McCormick envelopes on each subregion. This framework enables tighter convex relaxations and dual bounds in global optimization, particularly mixed-integer nonlinear programming (MINLP), polynomial optimization, and PDE-constrained problems with nonconvexities. Recent developments have introduced dynamic, sparse, and projection-based partitioning to efficiently balance relaxation strength and computational overhead, accompanied by rigorous bound-tightening and combinatorial encoding strategies.

1. Foundations: McCormick Relaxations and Motivation for Piecewise Refinement

Classical McCormick relaxations provide tightest convex polyhedral relaxations of a bilinear term $z = x_i x_j$ over a box $x_i \in [x_i^l, x_i^u]$, $x_j \in [x_j^l, x_j^u]$ through four facet-defining inequalities: $$\begin{aligned} z &\ge x_i^l x_j + x_j^l x_i - x_i^l x_j^l \ z &\ge x_i^u x_j + x_j^u x_i - x_i^u x_j^u \ z &\le x_i^l x_j + x_j^u x_i - x_i^l x_j^u \ z &\le x_i^u x_j + x_j^l x_i - x_i^u x_j^l \ \end{aligned}$$ For higher-degree terms $x_1 \cdots x_k$, relaxations are built recursively via auxiliary bilinears. However, if the variable bounds are wide, the McCormick envelope is excessively loose, which motivates mesh refinement. By partitioning each variable’s domain into subintervals and enforcing a separate McCormick envelope on each “piece,” the outer approximation becomes strictly tighter as the mesh is refined, converging to the exact nonconvex set as the partition becomes arbitrarily fine (Nagarajan et al., 2016, He et al., 2023).

Uniform partitioning, wherein the domain is split with equal size subintervals, introduces $M^k$ subregions for $k$-linear terms, causing exponential scaling in binary variables and constraints. This combinatorial growth is often unnecessary since large portions of the solution domain may not contain any feasible or near-optimal solutions (Nagarajan et al., 2016).

2. Formulations and Methodologies for Piecewise McCormick Relaxations

2.1 Disjunctive and Mixed-Integer Encodings

Partitioning $[x_i^l, x_i^u]$ and $[x_j^l, x_j^u]$ into $M_i$, $M_j$ intervals, define binaries $y_{i,k}$, $y_{j,\ell}$ so exactly one interval per variable is active ($\sum_k y_{i,k}=1$ and $y_{i,k}\in\{0,1\}$). The piecewise McCormick envelope for $z = x_i x_j$ over these subrectangles is: $$\begin{aligned} z &\ge (x^l_i \cdot y_i) x_j + (x^l_j \cdot y_j) x_i - (x^l_i \cdot y_i)(x^l_j \cdot y_j) \ z &\ge (x^u_i \cdot y_i) x_j + (x^u_j \cdot y_j) x_i - (x^u_i \cdot y_i)(x^u_j \cdot y_j) \ z &\le (x^l_i \cdot y_i) x_j + (x^u_j \cdot y_j) x_i - (x^l_i \cdot y_i)(x^u_j \cdot y_j) \ z &\le (x^u_i \cdot y_i) x_j + (x^l_j \cdot y_j) x_i - (x^u_i \cdot y_i)(x^l_j \cdot y_j) \ \end{aligned}$$ where $x^l_i \cdot y_i = \sum_k x^l_{i,k} y_{i,k}$, and products $y_{i,k} y_{j,\ell}$ are encoded via standard linearization (Nagarajan et al., 2016, He et al., 2023). For higher-order (multilinear) or composite functions, the relaxation is extended recursively; specialized incremental encodings can further reduce the binary variable count (He et al., 2023).

2.2 Dynamic Partitioning and Adaptive Refinement

Dynamic multivariate partitioning refines partitions adaptively in “regions of interest” near incumbent or prospective solutions. At each outer iteration, only those subboxes containing the current MILP solution and having width above a parameter $\epsilon$ are refined. A scaling parameter $\Delta > 1$ controls granularity. This approach decouples the number of partitions from the original variable domain size and minimizes the number of introduced binaries (Nagarajan et al., 2016).

Constraint-Programming (CP) bound contraction is frequently deployed prior to partitioning. Solving MILPs to minimize/maximize each variable under an optimality cutoff contracts the feasible bounds substantially (reducing width by up to 99% in computational studies), which directly translates to sparser, localized, and more effective piecewise McCormick relaxations (Nagarajan et al., 2016).

2.3 Piecewise Relaxations in PDE-Constrained and Continuous Domains

In infinite-dimensional (PDE-constrained) settings, the domain is partitioned into $N$ cells of maximum diameter $h$, and the involved nonlinearity (e.g., $u\cdot w$) is approximated as the product of cellwise averages, introducing $N$ McCormick constraints per cell. A priori error estimates show the relaxation gap vanishes as $O(h^\alpha)$, and optimization-based bound tightening on cell variables further sharpens the convex relaxation (Leyffer et al., 2024).

3. Computational Hierarchies and Branch-and-Bound Integration

Piecewise McCormick relaxations are fundamental in spatial branch-and-bound frameworks for solving global MINLPs. For each nonconvex term, the relaxations are locally recomputed at each node and recursively refined by branching:

• For trilinear monomials $x y z$, distinct hierarchical relaxations exist: the convex hull (full), three “double-McCormick” (sequential bilinear) relaxations, and their respective volumes, all computable in closed form (Speakman et al., 2017).
• Volume-minimizing branching strategies determine the optimal variable and splitting point to minimize the sum of subregion volumes, proven to produce smaller search trees for multilinear programs (Speakman et al., 2017).
• Bound tightening and cut selection are performed globally and locally per branch, exploiting the refined piecewise structure to close relaxation gaps efficiently (Nagarajan et al., 2016, Müller et al., 2019).

4. Strengthened Separation and Propagation: Projections and Polyhedral Refinement

By projecting the feasible region onto the $(x, y)$-space, additional valid inequalities for each bilinear pair can be obtained via LP-based “shooting” experiments. The resulting local polytope $P$ (intersecting up to four box and up to four facet cuts) is used to further tighten McCormick envelopes:

• Locatelli’s closed-form tangency formulas yield the tightest possible tangent plane at each boundary or critical point, strengthening the relaxation beyond axis-parallel box cases (Müller et al., 2019).
• Bound tightening exploits the polyhedral $P$ by enumerating the finite set of vertices and interior critical points, ensuring extremal bounds for $z=xy$ are attained.
• These methods are integrated into solvers as term-wise data structures, with separation and bound-tightening steps executed in LP callbacks; cost is modest (typically a handful of extra LPs per term and a negligible number of additional constraints) (Müller et al., 2019).

5. Comparative Computational Efficacy and Parameter Sensitivities

Extensive computational studies on MINLPLIB and synthetic polynomial optimization instances highlight several points:

• Dynamic (DTMC) partitioning dramatically reduces the required binary variables (by 50–80%) for identical or better relaxation tightness compared to uniform partitioning (Nagarajan et al., 2016).
• On MINLP/PDE settings, piecewise McCormick relaxations provide a rigorous a priori relaxation gap determined by partition mesh size, with convergence guarantees to the true optimum as $h \to 0$ (Leyffer et al., 2024).
• Incremental combinatorial encodings (Δ-based MILP) achieve up to 60–70% closure of the root-node relaxation gap relative to base McCormick relaxations, with only a modest computational overhead (often just a few times slower than classical McCormick) (He et al., 2023).
• Key parameters include the partition fineness $\Delta$, minimum width $\epsilon$, and tightness/tolerance in bound contraction; their selection affects both MILP size and relaxation quality (Nagarajan et al., 2016).

6. Implementation Strategies and Practical Considerations

Implementation of piecewise McCormick relaxations combines:

• Maintaining per-variable lists of disjoint intervals and binaries, with bilinear products linearized efficiently only for active subbox pairs.
• Solver integration of dynamic partitioning and adaptive refinement mechanisms, including heuristics for early stopping, efficient memory management, and tailored MILP model builds (Nagarajan et al., 2016).
• Use of polyhedral projections for bilinear pairs, reusing them throughout the branch-and-bound tree unless box bounds change substantially, which reduces recomputation overhead (Müller et al., 2019).
• In PDE-constrained cases, leveraging structure for parallel bound-tightening and using direct or multigrid linear solvers in relaxation steps (Leyffer et al., 2024).

7. Convergence Guarantees and Limitations

Piecewise McCormick relaxations are mathematically consistent: as the partitions become arbitrarily fine, the convex relaxation converges to the nonconvex feasible region both in finite- and infinite-dimensional settings, with minimizers of the relaxations converging (in the $\Gamma$-limit sense) to the true optimal solution (Leyffer et al., 2024). Main limitations are the exponential scaling of uniform partitioning, potential combinatorial explosion for high-degree multilinears, and the computational cost of aggressive bound tightening. Dynamic and projection-based partitioning strategies efficiently address these concerns in practice (Nagarajan et al., 2016, Müller et al., 2019).