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

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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.

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Authors and Affiliations

  1. Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Thomas C. Sharkey

  2. Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI, 48109-2117, USA

    H. Edwin Romeijn

  3. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Joseph Geunes

Authors
  1. Thomas C. Sharkey
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  2. H. Edwin Romeijn
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  3. Joseph Geunes
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Correspondence to Thomas C. Sharkey.

Additional information

The work of T. C. Sharkey was supported by a National Science Foundation Graduate Research Fellowship. The work of H. E. Romeijn was supported by the National Science Foundation under grant no. DMI-0355533. The work of J. Geunes was supported by the National Science Foundation under grants no. DMI-0355533 and DMI-0322715.

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Sharkey, T.C., Romeijn, H.E. & Geunes, J. A class of nonlinear nonseparable continuous knapsack and multiple-choice knapsack problems. Math. Program. 126, 69–96 (2011). https://doi.org/10.1007/s10107-009-0274-9

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  • DOI: https://doi.org/10.1007/s10107-009-0274-9

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