IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v59y2011i5p1225-1232.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1110.0930
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1110.0930?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. F. Guerriero & P. Tseng, 2002. "Implementation and Test of Auction Methods for Solving Generalized Network Flow Problems with Separable Convex Cost," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 113-144, October.
    2. Ogbe, Emmanuel & Li, Xiang, 2017. "A new cross decomposition method for stochastic mixed-integer linear programming," European Journal of Operational Research, Elsevier, vol. 256(2), pages 487-499.
    3. Minghe Sun, 2005. "Warm-Start Routines for Solving Augmented Weighted Tchebycheff Network Programs in Multiple-Objective Network Programming," INFORMS Journal on Computing, INFORMS, vol. 17(4), pages 422-437, November.
    4. Festa, P. & Guerriero, F. & Laganà, D. & Musmanno, R., 2013. "Solving the shortest path tour problem," European Journal of Operational Research, Elsevier, vol. 230(3), pages 464-474.
    5. R. Fourer & H. Gassmann & J. Ma & R. Martin, 2009. "An XML-based schema for stochastic programs," Annals of Operations Research, Springer, vol. 166(1), pages 313-337, February.
    6. Mongeau, Marcel & Sartenaer, Annick, 1995. "Automatic decrease of the penalty parameter in exact penalty function methods," European Journal of Operational Research, Elsevier, vol. 83(3), pages 686-699, June.
    7. Moreno, Alfredo & Munari, Pedro & Alem, Douglas, 2019. "A branch-and-Benders-cut algorithm for the Crew Scheduling and Routing Problem in road restoration," European Journal of Operational Research, Elsevier, vol. 275(1), pages 16-34.
    8. P. Beraldi & F. Guerriero & R. Musmanno, 1997. "Efficient Parallel Algorithms for the Minimum Cost Flow Problem," Journal of Optimization Theory and Applications, Springer, vol. 95(3), pages 501-530, December.
    9. Sarah Root & Amy Cohn, 2008. "A novel modeling approach for express package carrier planning," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(7), pages 670-683, October.
    10. Dayal Madhukar & Verma, Sanjay, 2015. "Multi-processor Exact Procedures for Regular Measures of the Multi-mode RCPSP," IIMA Working Papers WP2015-03-25, Indian Institute of Management Ahmedabad, Research and Publication Department.
    11. Ahuja, Ravindra K., 1956- & Orlin, James B., 1953-, 1992. "Use of representative operation counts in computational testings of algorithms," Working papers 3459-92., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    12. X-Y Li & Y P Aneja & F Baki, 2010. "An ant colony optimization metaheuristic for single-path multicommodity network flow problems," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(9), pages 1340-1355, September.
    13. Herminia Calvete & Lourdes del-Pozo & José Iranzo, 2012. "Algorithms for the quickest path problem and the reliable quickest path problem," Computational Management Science, Springer, vol. 9(2), pages 255-272, May.
    14. Avinash Unnikrishnan & Varunraj Valsaraj & Steven Waller, 2009. "Stochastic and Dynamic Shipper Carrier Network Design Problem," Networks and Spatial Economics, Springer, vol. 9(4), pages 525-550, December.
    15. Francesca Guerriero & Roberto Musmanno & Valerio Lacagnina & Antonio Pecorella, 2001. "A Class of Label-Correcting Methods for the K Shortest Paths Problem," Operations Research, INFORMS, vol. 49(3), pages 423-429, June.
    16. Claudio Gambella & Joe Naoum-Sawaya & Bissan Ghaddar, 2018. "The Vehicle Routing Problem with Floating Targets: Formulation and Solution Approaches," INFORMS Journal on Computing, INFORMS, vol. 30(3), pages 554-569, August.
    17. Sun, Minghe & Aronson, Jay E. & McKeown, Patrick G. & Drinka, Dennis, 1998. "A tabu search heuristic procedure for the fixed charge transportation problem," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 441-456, April.
    18. Mingue SUn, 2010. "A Branch-and-Bound Algorithm for Representative Integer Efficient Solutions in Multiple Objective Network Programming Problems," Working Papers 0007, College of Business, University of Texas at San Antonio.
    19. Tonbari, Mohamed El & Ahmed, Shabbir, 2023. "Consensus-based Dantzig-Wolfe decomposition," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1441-1456.
    20. Marins, Fernando A. S. & Senne, Edson L. F. & Darby-Dowman, Ken & Machado, Arlene F. & Perin, Clovis, 1997. "Algorithms for network piecewise-linear programs: A comparative study," European Journal of Operational Research, Elsevier, vol. 97(1), pages 183-199, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:59:y:2011:i:5:p:1225-1232. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.