Combinatorial separation algorithms for the continuous knapsack polyhedra with divisible capacities

TL;DR

This paper develops efficient combinatorial separation algorithms for two types of continuous knapsack polyhedra with divisible capacities, enhancing the computational tools for mixed integer programming.

Contribution

It introduces polynomial-time combinatorial separation algorithms for the continuous ≥- and ≤-knapsack polyhedra, including new inequalities that fully describe these polyhedra.

Findings

Separation algorithms operate in O(nm + m log m) time.

Polynomial-time separation for the family of ≥-partition inequalities.

Complete polyhedral description for the ≤-knapsack polyhedron with new inequalities.

Abstract

It is important to design separation algorithms of low computational complexity in mixed integer programming. We study the separation problems of the two continuous knapsack polyhedra with divisible capacities. The two polyhedra are the convex hulls of the sets which consist of $ n $ nonnegative integer variables, one unbounded continuous, $ m $ bounded continuous variables, and one linear constraint in either $ \geq $ or $ \leq $ form where the coefficients of integer variables are integer and divisible. Wolsey and Yaman (Math Program 156: 1--20, 2016) have shown that the polyhedra can be described by adding the two closely related families of partition inequalities. However, no combinatorial separation algorithm is known yet for the polyhedra. In this paper, for each polyhedron, we provide a combinatorial separation algorithm with the complexity of $ \mathcal{O}(nm+m\log m ) $. In particular, we solve the separation problem for the continuous $ \geq $-knapsack polyhedron by presenting a polynomial-time separation algorithm for the family of $ \geq $-partition inequalities. For the continuous $ \leq $-knapsack polyhedron, we derive the complemented partition inequalities, which dominate the $ \leq $-partition inequalities. The continuous $ \leq $-knapsack polyhedron is completely described with the addition of the proposed inequalities. We again show that the family of the proposed inequalities can be separated in polynomial time.