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Deepest Cuts for Benders Decomposition

Mojtaba Hosseini, John Turner

Abstract

Since its inception, Benders Decomposition (BD) has been successfully applied to a wide range of large-scale mixed-integer (linear) problems. The key element of BD is the derivation of Benders cuts, which are often not unique. In this paper, we introduce a novel unifying Benders cut selection technique based on a geometric interpretation of cut ``depth'', produce deepest Benders cuts based on $\ell_p$-norms, and study their properties. Specifically, we show that deepest cuts resolve infeasibility through minimal deviation (in a distance sense) from the incumbent point, are relatively sparse, and may produce optimality cuts even when classical Benders would require a feasibility cut. Leveraging the duality between separation and projection, we develop a Guided Projections Algorithm for producing deepest cuts while exploiting the combinatorial structure and decomposability of problem instances. We then propose a generalization of our Benders separation problem, which not only brings several well-known cut selection strategies under one umbrella, but also, when endowed with a homogeneous function, enjoys several properties of geometric separation problems. We show that, when the homogeneous function is linear, the separation problem takes the form of the Minimal Infeasible Subsystems (MIS) problem. As such, we provide systematic ways of selecting the normalization coefficients of the MIS method, and introduce a Directed Depth-Maximizing Algorithm for deriving these cuts. Inspired by the geometric interpretation of distance-based cuts and the repetitive nature of two-stage stochastic programs, we introduce a tailored algorithm to further facilitate deriving these cuts. Our computational experiments on various benchmark problems illustrate effectiveness of deepest cuts in reducing both computation time and number of Benders iterations, and producing high quality bounds at early iterations.

Deepest Cuts for Benders Decomposition

Abstract

Since its inception, Benders Decomposition (BD) has been successfully applied to a wide range of large-scale mixed-integer (linear) problems. The key element of BD is the derivation of Benders cuts, which are often not unique. In this paper, we introduce a novel unifying Benders cut selection technique based on a geometric interpretation of cut ``depth'', produce deepest Benders cuts based on -norms, and study their properties. Specifically, we show that deepest cuts resolve infeasibility through minimal deviation (in a distance sense) from the incumbent point, are relatively sparse, and may produce optimality cuts even when classical Benders would require a feasibility cut. Leveraging the duality between separation and projection, we develop a Guided Projections Algorithm for producing deepest cuts while exploiting the combinatorial structure and decomposability of problem instances. We then propose a generalization of our Benders separation problem, which not only brings several well-known cut selection strategies under one umbrella, but also, when endowed with a homogeneous function, enjoys several properties of geometric separation problems. We show that, when the homogeneous function is linear, the separation problem takes the form of the Minimal Infeasible Subsystems (MIS) problem. As such, we provide systematic ways of selecting the normalization coefficients of the MIS method, and introduce a Directed Depth-Maximizing Algorithm for deriving these cuts. Inspired by the geometric interpretation of distance-based cuts and the repetitive nature of two-stage stochastic programs, we introduce a tailored algorithm to further facilitate deriving these cuts. Our computational experiments on various benchmark problems illustrate effectiveness of deepest cuts in reducing both computation time and number of Benders iterations, and producing high quality bounds at early iterations.

Paper Structure

This paper contains 71 sections, 30 theorems, 110 equations, 11 figures, 9 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Classical Benders Decomposition
  3. A Unifying Decomposition Scheme
  4. Deepest Benders Cuts
  5. Euclidean Deepest Cuts
  6. Lp-norm Deepest Cuts
  7. A Primal Projection Perspective of the Separation Problem
  8. General Benders Distance Functions
  9. Normalized Distance Functions and Normalization Functions
  10. Projective Normalization Functions and Projective Distance Functions.
  11. Linear Pseudonorms
  12. Classical Benders pseudonorm.
  13. Relaxed L1 pseudonorm.
  14. Magnanti-Wong-Papadakos pseudonorm.
  15. Conforti-Wolsey pseudonorm.
  16. ...and 56 more sections

Key Result

Proposition 1

Given $q\ge 1$ and $p\ge 1$ such that $\ell_p$ is the dual norm of $\ell_q$ (i.e., $\frac{1}{p}+\frac{1}{q}=1$), the $\ell_q$-distance from the point $\hat{ \boldsymbol{z} }\in\mathbb{R}^{n+1}$ to hyperplane ${\BFalpha}^{\top}{ \boldsymbol{z} }+\beta=0$ is

Figures (11)

  • Figure 1: Deepest (red) versus classical (blue) Benders cut selection. Double-crossed segments correspond to Euclidean distances from $(\hat{ \boldsymbol{y} },\hat{\eta})$ to candidate cuts.
  • Figure 2: Primal-dual perspectives of the separation problem. Separating $(\hat{ \boldsymbol{y} },\hat{\eta})\notin \mathcal{E}$ with an $\ell_p$-deepest cut accounts for finding $(\tilde{ \boldsymbol{y} },\tilde{\eta})\in\mathcal{E}$ with minimum $\ell_q$-distance to $(\hat{ \boldsymbol{y} },\hat{\eta})$. The red lines illustrate the contour lines of the objective value of SSP, which also correspond to $\ell_q$-balls around $(\hat{ \boldsymbol{y} },\hat{\eta})$.
  • Figure 3: Primal view of projective normalization function $g$, and $(\tilde{ \boldsymbol{y} },\tilde{\eta})\in \mathcal{E}$, the $g$-projection of $(\hat{ \boldsymbol{y} },\hat{\eta})\notin \mathcal{E}$.
  • Figure 4: Comparing the performance of Benders with different cut selection strategies and Cplex (MILP) on large instances of CFLP and UFLP.
  • Figure 5: Comparing performance of $\ell_1$-deepest cuts, CW cuts and CB cuts on instances of stochastic CFLP (top), stochastic MCNDP (middle), and SNIP (bottom).
  • ...and 6 more figures

Theorems & Definitions (45)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Theorem \oldthetheorem
  • Corollary 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Definition 1: Normalized distance function
  • ...and 35 more