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A New Combinatorial Algorithm for Separable Convex Resource Allocation with Nested Bound Constraints

Author

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  • Zeyang Wu

    (Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, Minnesota 55455)

  • Kameng Nip

    (School of Mathematical Sciences, Xiamen University, Xiamen 361005, China)

  • Qie He

    (Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, Minnesota 55455)

Abstract

The separable convex resource allocation problem with nested bound constraints aims to allocate B units of resources to n activities to minimize a separable convex cost function, with lower and upper bounds on the total amount of resources that can be consumed by nested subsets of activities. We develop a new combinatorial algorithm to solve this model exactly. Our algorithm is capable of solving instances with millions of activities in several minutes. The running time of our algorithm is at most 73% of the running time of the current best algorithm for benchmark instances with three classes of convex objectives. The efficiency of our algorithm derives from a combination of constraint relaxation and divide and conquer based on infeasibility information. In particular, nested bound constraints are relaxed first; if the solution obtained violates some bound constraints, we show that the problem can be divided into two subproblems of the same structure and smaller sizes according to the bound constraint with the largest violation. Summary of Contribution. The resource allocation problem is a collection of optimization models with a wide range of applications in production planning, logistics, portfolio management, telecommunications, statistical surveys, and machine learning. This paper studies the resource allocation model with prescribed lower and upper bounds on the total amount of resources consumed by nested subsets of activities. These nested bound constraints are motivated by storage limits, time-window requirements, and budget constraints in various applications. The model also appears as a subproblem in models for green logistics and machine learning, and it has to be solved repeatedly. The model belongs to the class of computationally challenging convex mixed-integer nonlinear programs. We develop a combinatorial algorithm to solve this model exactly. Our algorithm is faster than the algorithm that currently has the best theoretical complexity in the literature on an extensive set of test instances. The efficiency of our algorithm derives from the combination of an infeasibility-guided divide-and-conquer framework and a scaling-based greedy subroutine for resource allocation with submodular constraints. This paper also showcases the prevalent mismatch between the theoretical worst-case time complexity of an algorithm and its practical efficiency. We have offered some explanations of this mismatch through the perspectives of worst-case analysis, specially designed instances, and statistical metrics of numerical experiments. The implementation of our algorithm is available on an online repository.

Suggested Citation

  • Zeyang Wu & Kameng Nip & Qie He, 2021. "A New Combinatorial Algorithm for Separable Convex Resource Allocation with Nested Bound Constraints," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1197-1212, July.
    • Handle: RePEc:inm:orijoc:v:33:y:2021:i:3:p:1197-1212
    DOI: 10.1287/ijoc.2020.1006
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    References listed on IDEAS

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    1. Dorit S. Hochbaum, 1994. "Lower and Upper Bounds for the Allocation Problem and Other Nonlinear Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 390-409, May.
    2. Bernard O. Koopman, 1953. "The Optimum Distribution of Effort," Operations Research, INFORMS, vol. 1(2), pages 52-63, February.
    3. Patriksson, Michael, 2008. "A survey on the continuous nonlinear resource allocation problem," European Journal of Operational Research, Elsevier, vol. 185(1), pages 1-46, February.
    4. Patriksson, Michael & Strömberg, Christoffer, 2015. "Algorithms for the continuous nonlinear resource allocation problem—New implementations and numerical studies," European Journal of Operational Research, Elsevier, vol. 243(3), pages 703-722.
    5. Jan Kronqvist & Andreas Lundell & Tapio Westerlund, 2018. "Reformulations for utilizing separability when solving convex MINLP problems," Journal of Global Optimization, Springer, vol. 71(3), pages 571-592, July.
    6. Thijs Klauw & Marco E. T. Gerards & Johann L. Hurink, 2017. "Resource allocation problems in decentralized energy management," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 39(3), pages 749-773, July.
    7. Ricardo Fukasawa & Qie He & Fernando Santos & Yongjia Song, 2018. "A Joint Vehicle Routing and Speed Optimization Problem," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 694-709, November.
    8. Groenevelt, H., 1991. "Two algorithms for maximizing a separable concave function over a polymatroid feasible region," European Journal of Operational Research, Elsevier, vol. 54(2), pages 227-236, September.
    9. Awi Federgruen & Henry Groenevelt, 1986. "Optimal Flows in Networks with Multiple Sources and Sinks, with Applications to Oil and Gas Lease Investment Programs," Operations Research, INFORMS, vol. 34(2), pages 218-225, April.
    10. Foldes, Stephan & Soumis, Francois, 1993. "PERT and crashing revisited: Mathematical generalizations," European Journal of Operational Research, Elsevier, vol. 64(2), pages 286-294, January.
    11. He, Qie & Zhang, Xiaochen & Nip, Kameng, 2017. "Speed optimization over a path with heterogeneous arc costs," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 198-214.
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