An efficient hybrid heuristic method for the 0-1 exact k-item quadratic knapsack problem

Abstract

ABSTRACT. The 0-1 exact k-item quadratic knapsack problem (E − k QK P) consists of maximizing a quadratic function subject to two linear constraints: the first one is the classical linear capacity constraint; the second one is an equality cardinality constraint on the number of items in the knapsack. Most instances of this NP-hard problem with more than forty variables cannot be solved within one hour by a commercial software such as CPLEX 12.1. We propose therefore a fast and efficient heuristic method which produces both good lower and upper bounds on the value of the problem in reasonable time. Specifically, it integrates a primal heuristic and a semidefinite programming reduction phase within a surrogate dual heuristic. A large computational experiments over randomly generated instances with up to 200 variables validates the relevance of the bounds produced by our hybrid dual heuristic, which yields known optima (and prove optimality) in 90% (resp. 76%) within 100 seconds on the average.

Exact Solution Methods for the k-Item Quadratic Knapsack Problem

“…All experiments have been computed on an Intel i7-2600 quad core 3.4 GHz with 8 GB of RAM, using only one core. The computational results have been obtained for randomly generated instances from [18] with up to 150 variables. The time limit for each approach is 3 hours.…”

“…As a heuristic method at the root node we chose the primal heuristic denoted by H pri in [18], which is an adaption of a well-known heuristic developed by Billionnet and Calmels [7] for the classical quadratic knapsack problem (QKP). This primal heuristic combines a greedy algorithm with local search.…”

“…At each node of the branch-and-bound tree, we apply a variable fixation heuristic inspired from H sdp [18]. This heuristic method uses the solution of the semidefinite relaxation obtained at each node, it fixes variables under some treshold ǫ > 0 to zero and applies the primal heuristic over the reduced problem.…”

An Improved Convex 0-1 Quadratic Program Reformulation for Chance-Constrained Quadratic Knapsack Problems

“…In particular, we consider here the following k-item probabilistic QKP [see (Létocart et al, 2010)]: In particular, we consider here the following k-item probabilistic QKP [see (Létocart et al, 2010)]:…”

“…As in Billionnet et al (2009) and Létocart et al (2010), the objective function in (kCQKP) can be rewritten as As in Billionnet et al (2009) and Létocart et al (2010), the objective function in (kCQKP) can be rewritten as…”