Approximating value functions via corner Benders' cuts

TL;DR

This work introduces corner Benders' cuts, a novel method to harvest subproblem structure by projecting a corner relaxation of the second-stage polyhedron into the master problem space. By leveraging basis information and reverse-polar separation, the approach generates multiple facet-defining cuts that strengthen the epigraph of the value function and often accelerate convergence. The method naturally recovers Dantzig–Wolfe bounds and, when specialized to network-flow structures, yields strong root bounds and competitive performance on the VRPSD, outperforming several baselines in many instances. The results demonstrate the practical impact of exploiting corner relaxations for Benders' cuts, suggesting broad applicability to large-scale two-stageLP/MIP problems with network-flow substructures and stochastic recourse.

Abstract

We introduce a novel technique to generate Benders' cuts from a conic relaxation ("corner") derived from a basis of a higher-dimensional polyhedron that we aim to outer approximate in a lower-dimensional space. To generate facet-defining inequalities for the epigraph associated to this corner, we develop a computationally-efficient algorithm based on a compact reverse polar formulation and a row generation scheme that handles the redundant inequalities. Via a known connection between arc-flow and path-flow formulations, we show that our method can recover the linear programming bound of a Dantzig-Wolfe formulation using multiple cuts in the projected space. In computational experiments, our generic technique enhances the performance of a problem-specific state-of-the-art algorithm for the vehicle routing problem with stochastic demands, a well-studied variant of the classic capacitated vehicle routing problem that accounts for customer demand uncertainty.

Figures (10)

• Figure 1: The basis associated to a vertex $y^*$ of $Y$ yields the translated cone (the relaxed corner polyhedron) $C \supseteq Y$, shown in the left panel. The facets of $C$ can then be projected into the $(x, \theta)$-domain, depicted in the right panel.
• Figure 2: Example of a VRPSD instance with $V_+ = \{1, 2, 3\}$, $k = 2$, and $C = 3$. The edge costs are $c_{01} = c_{03} = c_{12} = c_{23} = 14$ and $c_{13} = c_{02} = 20$. The expected demand values $\bar{q}_v$ are given beside each customer $v \in V_+$. In this example, each $q_i$ follows a normal probability distribution with variance $0.1$, so if a q-route $R$ has $\bar{q}(R) \leq 2$, then $\mathbb{E}[\mathcal{Q}(R)] \approx 0$. The solution in (a) has total cost $\sum_{i \in [2]} (c(R_i) + \mathbb{E}[\mathcal{Q}(R_i)]) = 76 + \mathbb{E}[\mathcal{Q}((2, 1))] = 76 + (1/2) \cdot 28 = 90$, where the 1/2 appears because the normal distribution is symmetric and results in 50% chance of exceeding capacity at customer 1.
• Figure 3: Network $\mathcal{N} = (\mathcal{V}, \mathcal{A})$ corresponding to the example shown in Figure \ref{['figure:vrpsd']}. The numbers next to the arcs indicate the arc cost $d_a$. We omit these numbers when the arc cost is zero. From left to right, each layer $\ell$ (indexed by the second number of each node label, corresponding to accumulated demand) contains a node for every customer with $\bar{q}_v \le \ell$.
• Figure 4: Illustration of the ray (cycle) $r^{([3, 3], t)}$. Notice that $(\alpha^1)^\mathsf{ T} (Q r^{([3, 3], t)}) + d^\mathsf{ T} r^{([3, 3], t)}) = 0$.
• Figure 5: Computational results for instances of Parada et al.

Theorems & Definitions (27)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 1
• proof
• Remark 1
• Definition 7