Improved Benders Decomposition

• Improved Benders Decomposition is a unified framework that bridges Farkas-based and separation-based cut generation using alternative polyhedron and reverse polar sets.
• It utilizes a parameterized, LP-based cut-generation approach to yield supporting, Pareto-optimal, and facet-defining cuts, enhancing algorithmic efficiency.
• Empirical results demonstrate 2–3× speedups in challenging, degenerate problem instances while ensuring robust finite convergence guarantees.

Improved Benders Decomposition denotes a collection of recent theoretical and algorithmic advances that significantly enhance the practical and theoretical performance of classical Benders decomposition. These improvements address long-standing challenges in cut selection, support/facet-defining property, scenario aggregation, subproblem acceleration, solution scalability, and convergence guarantees for large-scale and degenerate problems. Central to this development is the unifying geometric perspective that connects the alternative polyhedron, reverse polar sets, and separation-based approaches, leading to new unified cut-generation procedures with strong polyhedral properties and provable efficiency improvements (Brandenberg et al., 2019).

1. Geometric Foundations: Alternative Polyhedron and Reverse Polar Set

The modern theory of improved Benders decomposition is built upon precise geometric objects:

• Alternative Polyhedron: For a master–subproblem pair at candidate point $(x^*,\eta^*)$, the (relaxed) alternative polyhedron is

$$P^\le(x^*,\eta^*) := \{(\gamma,\gamma_0)\in\R^m_+\times\R_+ \mid \gamma^\top A+\gamma_0 d^\top=0,\; \gamma^\top(b-Hx^*)+\gamma_0\eta^*\le-1\}.$$

The original version (FSZ10) differs only in replacing "$\leq -1$" with "= -1".

• Reverse Polar Set: For a closed convex set $C\subseteq\mathbb{R}^n$ and reference point $\bar{z}\notin C$, the reverse polar of $C-\bar{z}$ is

$$(C-\bar{z})^- := \{\pi\in\R^n \mid \pi^\top(x-\bar{z})\le -1 \;\forall\, x\in C\}.$$

In Benders applications, this set describes the collection of valid separating directions for the epigraph of the value function $z$.

The crucial insight is that $P^\le(x^*, \eta^*)$ forms an extended formulation for the reverse polar set of the epigraph. This correspondence underpins the unification of Farkas-based (alternative polyhedron) and separation-based (reverse polar) cut generation (Brandenberg et al., 2019).

2. Unified Cut Generation and Modified FSZ10 Scheme

A central shortcoming of prior cut selection—e.g., the FSZ10 criterion, choosing Farkas rays minimizing $\|\gamma\|_1+\gamma_0$—is that such cuts may not support the epigraph and can be weak, especially in degenerate or redundant cases.

The improved method introduces a parameterized objective of the form $(H\omega, -\omega_0)$ (i.e., requiring the cut direction $(\tilde\omega,\tilde\omega_0)$ to lie in the column space of $(H, -1)$). This parameterization achieves two key advances:

• Guaranteed Support: Solving the LP

$$\max\left\{\tilde\omega^\top\gamma+\tilde\omega_0\gamma_0 \mid (\gamma,\gamma_0) \in P^\le(x^*,\eta^*)\right\}$$

produces a cut that is always supporting, i.e., truly separates $(x^*,\eta^*)$ from the epigraph.

• Facet-Defining Cuts: For almost all choices of $(\omega,\omega_0)$ in the conic hull of the shifted epigraph, the resulting cut defines a facet (i.e., is maximal) of the epigraph polyhedron; only a measure-zero subset of parameters yields degenerate solutions.

Moreover, this approach produces a single LP-based algorithm for cut generation that unifies all classical approaches: MIS-cuts, Pareto-optimal cuts, and facet-defining cuts become special cases associated with specific choices of the cut-generating objective (Brandenberg et al., 2019).

3. Pareto-Optimal and Facet-Defining Cuts: Parameterization and Inclusion of CW18

The question of how to consistently select the strongest possible cuts is addressed by mapping different geometric or efficiency desiderata to regions in the parameter space:

• Pareto-Optimal Cuts (Magnanti–Wong): If $(\omega,\omega_0)$ is chosen in the relative interior of the convex hull of the shifted epigraph, solving the reverse polar LP returns cuts that are Pareto-optimal—undominated by any other valid cut on the relevant scenario set.
• Facet Cuts: If $(\omega,\omega_0)$ lies in the conic hull of the shifted epigraph, the resulting extremal cut corresponds to a facet of the epigraph, with uniqueness except on lower-dimensional faces.

The method of Conforti and Wolsey (CW18), which arises by taking an interior point of the epigraph and maximizing the distance in that direction, is revealed as a special case within this unified framework. Thus, the improved Benders decomposition subsumes both FSZ10/Farkas-based and CW18-type separation (Brandenberg et al., 2019).

Cut Type	Parameter Region	Key Property
Supporting	Any $(\omega,\omega_0)$ in conic hull	Always separates epigraph at current point
Facet-defining	Generic $(\omega,\omega_0)$ in conic hull	Maximal, usually unique cuts
Pareto-optimal	$(\omega,\omega_0)$ in relative interior (conv)	Undominated over relevant scenario region

4. Algorithmic Implementation and Convergence Properties

The improved cut-generation scheme admits a compact, fully explicit LP-based algorithm:

• CutGeneration$(x^*,\eta^*,\omega,\omega_0)$: Solve the LP described above; extract $\pi = H^\top\gamma^*$, $\pi_0 = -\gamma_0^*$, and $\alpha = \gamma^{*\top}b$, and return the cut $\pi^\top x + \pi_0\eta \le \alpha$.
• Integration in Benders Loop: At each iteration, solve the master, generate a cut using the improved criterion, and add it if violated.

Complexity is polynomial, as the LPs involved are of size $O(\#$ constraints $+\#$ coupling rows$)$. The number of Benders iterations is bounded above by the number of facets of the polyhedral epigraph restricted to the scenario set, ensuring finite convergence for polyhedral problems. Thus, improved Benders decomposition inherits the finite convergence guarantees of classical cutting-plane methods, but with stronger per-iteration progress (Brandenberg et al., 2019).

5. Computational Impact and Case Studies

On challenging instances such as multi-commodity network design, deployment of the improved cut selection achieves empirically observed 2–3× speedup over standard Benders decomposition, with more pronounced gains for hard, degenerate problems where vanilla methods may timeout. The generated cuts more efficiently close optimality gaps and reduce iteration counts (Brandenberg et al., 2019).

The algorithm retains all desirable computational properties: each cut is provably supporting (and, with probability one, facet-defining), master problems are not enlarged beyond standard Benders, and overall solution time and memory footprint are unaffected except for the reduced number of iterations.

6. Theoretical and Practical Significance

The improved Benders decomposition provides a single unifying framework that bridges the gap between the various classical notions of Benders cuts (weak/strong, supported/facet, dominated/Pareto-optimal). By introducing a parametrized family of cut-selection LPs—grounded in the relation between the alternative polyhedron and the reverse polar set—it enables fine control over the strength and polyhedral properties of generated cuts. The connection of FSZ10, Magnanti–Wong, and CW18 within this paradigm leads to clearer guidance for practitioners on cut selection and algorithm design.

In summary, the improved Benders decomposition paradigm:

• Bridges Farkas-based and separation-based cut generation as two faces of the same geometric structure.
• Provides a parameterized, LP-based mechanism to control the strength (support/facet/Pareto) of Benders cuts.
• Unifies previously distinct state-of-the-art approaches into a general and computationally feasible framework, with strong convergence guarantees and practical acceleration (Brandenberg et al., 2019).