Lifting of probabilistic cover inequalities

Abstract

We consider a chance-constrained binary knapsack problem where weights of items are independent and normally distributed. Probabilistic cover inequalities can be defined for the problem. The lifting problem for probabilistic cover inequalities is NP-hard. We propose a polynomial time approximate lifting method for probabilistic cover inequalities based on the robust optimization approach. We present computational experiments on multidimensional chance-constrained knapsack problems. The results show that our lifting method reduces the computation time substantially.

Introduction

We consider a binary knapsack problem with uncertain weights of items. The binary knapsack problem is to select items to maximize the sum of profits while satisfying knapsack capacity constraint. The binary knapsack problem without uncertainty is known to be NP-hard [13]. Many practical problems can be formulated using the binary knapsack constraint, and studies on the knapsack solution set can be applied to any mixed integer programming problems containing the knapsack constraint. Therefore the binary knapsack problem has been extensively studied [13], [17], and much research is focused on the polyhedral aspects of the binary knapsack solution set.

In applications of optimization, an important issue is how to deal with data uncertainty. Representative approaches to deal with data uncertainty are stochastic programming and robust optimization. Stochastic optimization considers a probabilistic distribution of uncertain data [6], and chance-constrained programming [8] is one of the useful approaches of stochastic programming. It finds the best solution satisfying constraints with probability at least a given threshold. The chance-constrained knapsack problem has been studied by many authors. Goel et al. [9] proposed a polynomial time approximation scheme (PTAS) where weight of each item has a Poisson or exponential distribution. Kleinberg et al. [14] assumed that each weight has a Bernoulli distribution and presented an approximation algorithm. Klopfenstein et al. [15] proposed an approximation algorithm using robust optimization techniques. Goyal et al. [10] presented a PTAS using a parametric LP reformulation when item sizes are independent and normally distributed. Han et al. [12] proposed an efficient pseudopolynomial algorithm based on the robust optimization approach for finding a good upper bound on the optimal value when weight of each item has a normal distribution independent of the other items. This paper also focuses on the chance-constrained knapsack problem where weights are independent and normally distributed.

Cover inequalities are well-known valid inequalities for the binary knapsack solution set. Commercial optimization softwares such as CPLEX and Xpress-MP use cover inequalities in their branch-and-cut algorithms. Lifting is a powerful technique to strengthen cover inequalities [4], [21]. Cover inequalities also can be defined for the knapsack problems considering data uncertainty. Klopfenstein et al. [16] defined robust cover inequalities for the robust knapsack problem that is formulated using the uncertainty set of Bertsimas et al. [5]. Song et al. [20] considered the chance-constrained binary packing problem and proposed a problem formulation using probabilistic covers. They focused on the case with finite number of scenarios and considered an approximate lifting method. The chance-constrained knapsack problem is a special case of the chance-constrained packing problem. Atamtürk et al. [3] described cover inequalities for the submodular knapsack set and proposed a polynomial time approximate lifting algorithm using parametric LP. Note that the chance-constrained knapsack problem with independent and normally distributed weights is one of submodular knapsack problems. Atamtürk et al. [1] investigated separation and extension of cover inequalities for the conic quadratic knapsack constraint with generalized upper bound (GUB) constraints.

In this paper we propose a polynomial time approximate lifting algorithm for cover inequalities for the chance-constrained knapsack problem. We assume that weights of items are independent and normally distributed. We formulate the subproblem for lifting of probabilistic cover inequalities as a chance-constrained knapsack problem. Then we propose an approximate lifting algorithm using the efficient approximation algorithm of Han et al. [12].