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Projected Perspective Reformulations with Applications in Design Problems

Author

Listed:
  • Antonio Frangioni

    (Dipartimento di Informatica, Università di Pisa, Polo Universitario della Spezia, 19121 La Spezia, Italy)

  • Claudio Gentile

    (Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”---CNR, 00185 Rome, Italy)

  • Enrico Grande

    (Dipartimento di Ingegneria dell'Impresa, Università degli Studi di Roma “Tor Vergata,” 00133 Rome, Italy)

  • Andrea Pacifici

    (Dipartimento di Ingegneria dell'Impresa, Università degli Studi di Roma “Tor Vergata,” 00133 Rome, Italy)

Abstract

The perspective relaxation (PR) is a general approach for constructing tight approximations to mixed-integer nonlinear programs (MINLP) with semicontinuous variables. The PR of a MINLP can be formulated either as a mixed-integer second-order cone program (MI-SOCP), provided that the original objective function is SOCP-representable, or as a semi-infinite MINLP. In this paper, we show that under some further assumptions (rather restrictive, but satisfied in several practical applications), the PR of a mixed-integer quadratic program (MIQP) can also be reformulated as a piecewise-quadratic program (QP), ultimately yielding a QP relaxation of roughly the same size of the standard continuous relaxation. Furthermore, if the original problem has some exploitable structure, then this structure is typically preserved in the reformulation, thus allowing the construction of specialized approaches for solving the PR. We report on implementing these ideas on two MIQPs with appropriate structure: a sensor placement problem and a quadratic-cost (single-commodity) network design problem.

Suggested Citation

  • Antonio Frangioni & Claudio Gentile & Enrico Grande & Andrea Pacifici, 2011. "Projected Perspective Reformulations with Applications in Design Problems," Operations Research, INFORMS, vol. 59(5), pages 1225-1232, October.
    • Handle: RePEc:inm:oropre:v:59:y:2011:i:5:p:1225-1232
    DOI: 10.1287/opre.1110.0930
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    References listed on IDEAS

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    1. D. Klingman & A. Napier & J. Stutz, 1974. "NETGEN: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems," Management Science, INFORMS, vol. 20(5), pages 814-821, January.
    2. Antonio Frangioni, 2005. "About Lagrangian Methods in Integer Optimization," Annals of Operations Research, Springer, vol. 139(1), pages 163-193, October.
    3. Castro, J. & Nabona, N., 1996. "An implementation of linear and nonlinear multicommodity network flows," European Journal of Operational Research, Elsevier, vol. 92(1), pages 37-53, July.
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    Cited by:

    1. Jordi Castro & Antonio Frangioni & Claudio Gentile, 2014. "Perspective Reformulations of the CTA Problem with L 2 Distances," Operations Research, INFORMS, vol. 62(4), pages 891-909, August.
    2. Kevin C. Furman & Nicolas W. Sawaya & Ignacio E. Grossmann, 2020. "A computationally useful algebraic representation of nonlinear disjunctive convex sets using the perspective function," Computational Optimization and Applications, Springer, vol. 76(2), pages 589-614, June.
    3. Xiaojin Zheng & Yutong Pan & Zhaolin Hu, 2021. "Perspective Reformulations of Semicontinuous Quadratically Constrained Quadratic Programs," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 163-179, January.
    4. Antonio Frangioni & Fabio Furini & Claudio Gentile, 2016. "Approximated perspective relaxations: a project and lift approach," Computational Optimization and Applications, Springer, vol. 63(3), pages 705-735, April.

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