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Underestimation functions for a rank-two partitioning method

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  • Riccardo Cambini

    (University of Pisa)

Abstract

Low-rank problems are nothing but nonlinear minimization problems over polyhedrons where a linear transformation of the variables provides an objective function which actually depends on very few variables. These problems are often used in applications, for example, in concave quadratic minimization problems, multiobjective/bicriteria programs, location–allocation models, quantitative management science, data envelopment analysis, efficiency analysis and performance measurement. The aim of this paper is to deepen on the study of a solution method for a class of rank-two nonconvex problems having a polyhedral feasible region expressed by means of inequality/box constraints and an objective function of the kind $$\phi (c^Tx+c_0,d^Tx+d_0)$$ ϕ ( c T x + c 0 , d T x + d 0 ) . The rank-two structure of the problem allows to determine various localization conditions and underestimation functions. The stated theoretical conditions allow to determine a solution algorithm for the considered class of rank-two problems whose performance is witnessed by means of a deep computational test.

Suggested Citation

  • Riccardo Cambini, 2020. "Underestimation functions for a rank-two partitioning method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 465-489, December.
    • Handle: RePEc:spr:decfin:v:43:y:2020:i:2:d:10.1007_s10203-020-00288-6
    DOI: 10.1007/s10203-020-00288-6
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    References listed on IDEAS

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    1. Charnes, A. & Cooper, W. W. & Rhodes, E., 1978. "Measuring the efficiency of decision making units," European Journal of Operational Research, Elsevier, vol. 2(6), pages 429-444, November.
    2. Riccardo Cambini & Claudio Sodini, 2008. "A sequential method for a class of box constrained quadratic programming problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(2), pages 223-243, April.
    3. Alberto Cambini & Laura Martein, 2009. "Generalized Convexity and Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-70876-6, December.
    4. Riccardo Cambini & Claudio Sodini, 2002. "A Finite Algorithm for a Particular D.C. Quadratic Programming Problem," Annals of Operations Research, Springer, vol. 117(1), pages 33-49, November.
    5. Riccardo Cambini & Claudio Sodini, 2010. "Global optimization of a rank-two nonconvex program," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 165-180, February.
    6. Hoang Tuy, 2016. "Convex Analysis and Global Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-31484-6, August.
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    Cited by:

    1. Peiping Shen & Dianxiao Wu & Kaimin Wang, 2023. "Globally minimizing a class of linear multiplicative forms via simplicial branch-and-bound," Journal of Global Optimization, Springer, vol. 86(2), pages 303-321, June.

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