Exact Hull Reformulation for GDPs

• Exact hull reformulation for GDPs is a method that converts disjunctive and algebraic constraints into a continuous convex set that exactly matches the original feasible region.
• It eliminates epsilon approximations in nonlinear and quadratic constraints by multiplying through by binary variables, preserving the model's algebraic structure.
• This approach enhances MINLP solver performance by improving numerical stability, reducing branch-and-bound nodes, and ensuring reliable solution quality.

An exact hull reformulation for Generalized Disjunctive Programs (GDPs) refers to the process of transforming a GDP—comprising logical disjunctions combined with algebraic constraints—into an equivalent continuous relaxation whose feasible region precisely matches the convex hull of the original feasible set. Such reformulations yield tighter relaxations for Mixed-Integer Nonlinear Programming (MINLP) solvers, improving computational reliability, solution quality, and numerical stability. Recent advancements center on eliminating the need for approximation parameters in nonlinear and quadratic constraints, restoring algebraic structure and enhancing solver performance.

1. Core Concepts in Hull Reformulation for GDPs

Hull reformulation translates the discrete logical structure of a GDP into a continuous convex set while preserving all original constraints. In standard convex hull reformulation, for each disjunctive constraint indexed by $k$, the feasible region is partitioned into sets $S_{ik}$, each encoded by a binary variable $y_{ik}$ that signals activity. The continuous variables $x$ are split (“disaggregated”) into copies $v_{ik}$ for each disjunct, enforced by the reconstruction $x = \sum_{i \in D_k} v_{ik}$ and logical constraint $\sum_{i \in D_k} y_{ik} = 1$. Constraints are imposed on each copy so that the convex hull of the union of disjuncts is exactly described. The treatment of nonlinear constraints, especially quadratic ones, previously required relaxations or approximations, undermining tightness and accuracy.

2. Exact Reformulation of Quadratic Constraints

The recent exact hull reformulation for quadratically constrained GDPs (Gusev et al., 22 Aug 2025) eliminates the need for an $\epsilon$-approximation of the closure of the perspective function—a conventional workaround for division-by-zero when $y_{ik} = 0$. In standard methods, the constraint $h(x) = x^\top Q x + c^\top x + d \leq 0$ is reformulated for the $i$th disjunct as

$$\text{cl} \tilde{h}(v, y) = \begin{cases} \frac{v^\top Q v}{y} + c^\top v + d y, & y > 0 \ 0, & y = 0 \end{cases}$$

The new exact approach multiplies through by $y$, yielding an equivalent quadratic constraint: $v^\top Q v + (c^\top v) y + d y^2 \leq 0$ with the bounds $x^l y \leq v \leq x^u y$. This provides a formulation that matches the closure’s effect without introducing fractional terms or requiring a positive $\epsilon$.

3. Structural Properties and Advantages

Preservation of the original quadratic structure is central to this methodology. The reformulated problem remains a Mixed-Integer Quadratically Constrained Program (MIQCP), allowing direct exploitation by modern solvers. This tightness is mathematically exact—i.e., the feasible region of the continuous relaxation coincides with the convex hull of the original (disjunctive) feasible set. Solution bounds are not artificially weakened, and numerical stability is achieved by avoiding approximations that can produce degenerate or ill-posed constraints.

Advantages over conventional approaches include:

• Elimination of user-defined parameters ($\epsilon$) and associated relaxation errors.
• Exact equivalence between the original and reformulated constraints for both convex and nonconvex quadratics.
• Direct compatibility with MIQCP solvers, increasing solution reliability.

4. Applications and Computational Evidence

Comprehensive benchmarks have demonstrated computational improvements with the exact reformulation (Gusev et al., 22 Aug 2025):

• Random GDP Instances: Both convex and nonconvex quadratic GDPs display superior solution rates, fewer solver failures, and better solution times when using the exact approach (see cumulative performance plots).
• CSTR Network Optimization: The methodology efficiently resolves layout and configuration selection, outperforming $\epsilon$-approximated hull reformulations in both speed and error rates.
• k-Means Clustering: Exact hull reformulation yields more robust results, especially for solvers sensitive to numerical instability.
• Constrained Layout Problems: While some solver-dependent exceptions exist, overall solution quality and correctness are enhanced by eliminating approximation artifacts.

A detailed implication is that, for quadratic GDPs, minimizing the relaxation gap translates into fewer branch-and-bound nodes, more reliable bounds, and enhanced convergence rates for global solvers.

5. Extensions and Generalization

The paper suggests direct extension of the mathematical principle to higher-degree polynomials:

• For a general polynomial constraint $h(x) = \sum_k p_k(x)$ of degree $d$, the lifted constraint is $\sum_k p_k(v) y^{d-k} \leq 0$ in the reformulated space.
• This exact multiplication principle applies wherever perspective closures are involved, so rational expressions arising from division by $y$ can be regularized in the lifted space.

A plausible implication is that tight hull reformulations for complex, high-degree nonlinear GDPs can be systematically constructed, provided the algebraic structure is exploited as described.

6. Relation to Other Exact Hull Strategies

Complementary approaches—including conic GDP hull reformulation (Neira et al., 2021), convex hull for indicator constraints (2002.04681), and sharpness preservation in hybrid zonotopes via the reformulation-linearization technique (RLT) (Glunt et al., 21 Mar 2025)—demonstrate similar philosophies:

• Conic hull reformulations preserve second-order or exponential cone structures exactly, avoiding perspective approximations.
• RLT-based sharp hybrid zonotopes guarantee that convex relaxations match convex hulls, with explicit formulas quantifying complexity scaling.
• Both approaches emphasize that convex hull relaxation tightness directly improves solver efficiency and convergence.

This landscape of methodologies showcases a progression towards algebraic and geometric exactness in the convexification of GDPs, discouraging artificial relaxations and encouraging explicit exploitation of problem structure.

7. Future Directions

Prospective research avenues enabled by this exact reformulation framework include:

• API and solver integration that recognizes structure preservation, enabling more aggressive cut generation and bound tightening.
• Systematic paper of hybrid reformulation strategies (e.g., first convexifying nonconvex constraints, then hull reformulating).
• Testing at industrial scale, where complex GDPs with multiple nonlinear and logical layers demand both tightness and computational efficiency.
• Extension to disjunctive models with multilinear terms, applying similar exactification principles as have succeeded in the quadratic case.

This approach—centered on exact hull reformulation—thus stands out as a key mechanism for enhancing mixed-integer nonlinear optimization, especially in complex systems combining discrete and nonlinear constraints.