Bilinear Reformulation of Mixing Sets

• The paper introduces a bilinear reformulation for mixing sets in MIP, applying simplex-augmented spaces and knapsack constraints to model chance constraints.
• It presents a BLP procedure that aggregates bilinear terms and efficiently converts nonconvex regions into strong, valid linear inequalities.
• The approach unifies classical mixing inequalities with new convexification techniques, significantly improving convex hull approximation and computational performance.

The bilinear reformulation of mixing sets encompasses a sophisticated class of techniques in mixed-integer programming (MIP), especially as applied to chance-constrained programming. Mixing sets with knapsack constraints are central for modeling stochastic right-hand sides, and their polyhedral structure directly governs the tractability and strength of relaxation in formulations with mixing (bilinear) terms. The emergence of bilinear reformulations over simplex-augmented spaces and the evolution of convexification procedures have unified and generalized known families of valid inequalities, improved computational separation, and raised the standard of convex hull approximation for such sets.

1. Structural Foundations of Mixing Sets and Bilinear Terms

Mixing sets with knapsack constraints arise in MIP reformulations of chance-constrained programs over finite discrete distributions. The canonical form consists of a continuous variable $y$ and binary variables $z_j$ constrained by a knapsack-type summation. The constraint $y + h_j z_j \geq h_j$ (for scenario $j$) couples $y$ and $z_j$ bilinearly. Traditional studies have mapped many valid inequalities to pure knapsack polytopes, where only $z$-variables appear.

Recent results (Abdi et al., 2012) show that valid inequalities for the convex hull of the mixing set with a knapsack constraint can be written in a normalized form: $y + \sum_{j=1}^n a_j z_j \geq B$ with $a_j \geq 0$, the coefficient of $y$ normalized to unity, and $(a, B)$ contained in the coefficient polyhedron $G$, uniquely characterized by the scenario data $(h_1, ..., h_n)$, the knapsack coefficients, and specialized minimizer functions $f_k(a)$.

The mixing set thus serves as a key prototype for bilinear combinatorial structures, especially in modeling uncertainty under scenario selection.

2. Bilinear Reformulation over Lifted Simplex Spaces

A significant advance in the representation of mixing sets arises from their bilinear reformulation over simplex-augmented variable spaces (Davarnia et al., 17 Oct 2025). Specifically, the chance-constrained set is lifted into an extended space by introducing a vector $y$ lying in the binary simplex $\Delta_m$: $$\Delta_m = \left\{ y \in \mathbb{R}^m : y_j \in \{0,1\},\, \sum_{j=1}^m y_j = 1 \right\}$$ This augmentation yields an extended bilinear set

$$S = \{ (x, y)\ |\ x \in \Xi,\, y \in \Delta_m,\, y A^k x + b^k x + c^k y \geq d_k\ \forall k \in K \}$$

where bilinear products $x_i y_j$ encode scenario activation coupled with decision variables. The mixing set in the original $(z, x)$ variables is obtained as a projection of the union of sets $S(e^j)$ for unit vectors $e^j$ and the zero vector.

This bilinear reformulation leverages the structure of disjunctive programming and simplifies the nonconvex original feasible region into a union of easier-to-convexify bilinear sets parametrized by the simplex $y$.

3. Aggregation-Based Convexification and the BLP Procedure

To exploit the bilinear reformulation, a systematic convexification framework is introduced via the BLP (Bilinear Lift-and-Projection) method (Davarnia et al., 17 Oct 2025). The procedure, formalized as a sequence of aggregation steps, comprises:

• Selection (A1): Choose a base bilinear constraint (or disjunction).
• Aggregation (A2–A3): Form a weighted combination, applying nonnegative weights to each constraint, including knapsack constraints, to accumulate desired terms.
• Bilinear Term Management (R1–R2): Replace collections of bilinear terms $x_i y_j$ by maximizing positive coefficients for each index $i$ and projecting onto $x_i$. Similarly, $y$ terms are aggregated into a single constant term.
• Cancellation Condition (C1): Under a sufficient cancellation of bilinear and auxiliary $y$-terms (parametrized by dual variables and extremality), the aggregated inequality becomes linear in $(z, x)$; if the aggregation aligns with extreme dual points, the result is facet-defining for the convex hull.

This lifts, aggregates, and projects the polyhedral description into the original variable space, generating strong valid inequalities—generalizing mixing inequalities, strengthened star inequalities, and introducing new forms that are both theoretically and computationally dominant.

4. Characterization and Generalization of Valid Inequalities

The evolution from cardinality-constrained mixing sets (equal probability case) to general knapsack constraints is central (Abdi et al., 2012). For cardinality constraints, valid inequalities are frequently "star" inequalities of the form

$y + \sum_{j \in T} (h_j - h_{j+1}) z_j \geq h_1$

where $T$ is an index subset. In the generalized knapsack setting, arbitrary positive coefficients $a_j$ are permitted, and the inequality characterization expands to

$$y + \sum_{j=1}^n a_j z_j \geq B, \quad (a, B) \in G$$

with $G$ precisely described through scenario data and minimizer functions, capturing those inequalities that do not trivially arise from the knapsack polytope.

The latest convexification frameworks (Davarnia et al., 17 Oct 2025) synthesize these forms, yielding closed-form expressions for aggregated inequalities: $$z + \sum_{i \in P} (h_{t_i} - h_{t_{i+1}} + \delta_{t_i}) x_{t_i} + \sum_{j \in Q} \phi_{q_j} (1 - x_{q_j}) \geq h_{t_1}$$ with $P, Q$ denoting index subsets, and $\delta, \phi$ determined via explicit parameter formulas. When $\delta = 0$, earlier strengthened (star) inequalities are recovered as special cases.

A plausible implication is that this systematic approach unifies the polyhedral descriptions—allowing inequalities from classical, strengthened, and newly derived forms to emerge under one aggregation regime.

5. Efficient Separation Algorithms for Bilinear Cuts

The practical value of valid inequalities is contingent upon efficient separation—the identification of violated inequalities by fractional solutions during branch-and-cut or outer approximation algorithms. Earlier methods suffered from exponential candidate growth, especially as the scenario count increased.

Recent work (Abdi et al., 2012) establishes that, for the expanded family characterized via $G$, separation can be performed in polynomial time. By casting the separation problem as a linear program over the coefficient polyhedron (using the structure of minimizer functions), violated inequalities may be discovered without exhaustive enumeration: $$\text{Find } LP^* := \min_{(a, B) \in G} \left\{ \sum_{j=1}^n a_j z^*_j - B \right\}$$ and if $y^* + LP^* < 0$ at a candidate point $(y^*, z^*)$, the corresponding inequality is violated. This property is essential for practical large-scale computation.

Analogous advances in separating RLT (Reformulation-Linearization Technique) cuts for bilinear terms, including implicit products in MILP formulations, utilize row marking and projection filtering (Bestuzheva et al., 2022), which reduce separation time from over 50% to below 3% of total solver time for complex instances.

6. Computational Strength and Convex Hull Approximations

Computational experiments (Davarnia et al., 17 Oct 2025) evaluated the descriptive power of new inequalities on benchmarks, notably for mixing sets with cardinality constraints. Using complete convex hull enumeration, new aggregated inequalities (BL) captured over 90% of facet-defining inequalities for small to moderate sized sets (e.g., $m = 9, 10$), with gains of 15%–70% relative to the best known previous families.

In cases where the mixing set is equivalent to a pure star set, all approaches coincide; as the knapsack constraint becomes more descriptive (intermediate cardinality, general coefficients), the convexification approach and new inequalities provide substantial improvements in hull coverage and relaxation tightness.

A plausible implication is the shift in modeling philosophy—strong cuts are now explicitly derived via aggregation over extended simplex-bilinear spaces, rather than ad-hoc lifting or scenario-wise analysis.

7. Impact and Directions in Polyhedral Research

The technical advances in bilinear reformulation and convexification of mixing sets have multiple effects:

• Modeling: Enables systematic convexification of nonconvex MIP feasible regions in stochastic programming.
• Solver Integration: Polynomial separation and effective cut generation support branch-and-cut and RLT-based solvers in practice (Bestuzheva et al., 2022).
• Theoretical Unification: Aggregation-based frameworks provide a rigorous structure that subsumes previous families—strengthening, generalizing, and completing the polyhedral description for chance-constrained and probabilistic MIP settings.
• Future Research: The extension to complex stochastic models and higher-dimensional uncertainty sets is facilitated. The BL procedure offers extensible routes to derive strong cuts in other combinatorial nonconvex settings.

The contemporary view is that polyhedral analysis of mixing sets, historically dominated by scenario-wise mixture and cardinality, now rests on bilinear reformulation over extended spaces and methodical convexification through aggregation—a paradigm substantiated by both theoretical and computational evidence.