Mathematical Programming

, Volume 126, Issue 1, pp 69–96 | Cite as

A class of nonlinear nonseparable continuous knapsack and multiple-choice knapsack problems

  • Thomas C. Sharkey
  • H. Edwin Romeijn
  • Joseph Geunes
FULL LENGTH PAPER Series A

Abstract

This paper considers a general class of continuous, nonlinear, and nonseparable knapsack problems, special cases of which arise in numerous operations and financial contexts. We develop important properties of optimal solutions for this problem class, based on the properties of a closely related class of linear programs. Using these properties, we provide a solution method that runs in polynomial time in the number of decision variables, while also depending on the time required to solve a particular one-dimensional optimization problem. Thus, for the many applications in which this one-dimensional function is reasonably well behaved (e.g., unimodal), the resulting algorithm runs in polynomial time. We next develop a related solution approach to a class of continuous, nonlinear, and nonseparable multiple-choice knapsack problems. This algorithm runs in polynomial time in both the number of variables and the number of variants per item, while again dependent on the complexity of the same one-dimensional optimization problem as for the knapsack problem. Computational testing demonstrates the power of the proposed algorithms over a commercial global optimization software package.

Mathematics Subject Classification (2000)

90B30 90C26 90C30 90C46 

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Copyright information

© Springer and Mathematical Programming Society 2009

Authors and Affiliations

  • Thomas C. Sharkey
    • 1
  • H. Edwin Romeijn
    • 2
  • Joseph Geunes
    • 3
  1. 1.Department of Decision Sciences and Engineering SystemsRensselaer Polytechnic InstituteTroyUSA
  2. 2.Department of Industrial and Operations EngineeringThe University of MichiganAnn ArborUSA
  3. 3.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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