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Perturbation Approach to Sensitivity Analysis in Mathematical Programming

Author

Listed:
  • E. Castillo

    (University of Cantabria)

  • A. J. Conejo

    (University of Castilla-La Mancha)

  • C. Castillo

    (University of Granada)

  • R. Mínguez

    (University of Castilla-La Mancha
    Cornell University)

  • D. Ortigosa

    (University of La Rioja)

Abstract

This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush–Kuhn–Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn.

Suggested Citation

  • E. Castillo & A. J. Conejo & C. Castillo & R. Mínguez & D. Ortigosa, 2006. "Perturbation Approach to Sensitivity Analysis in Mathematical Programming," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 49-74, January.
    • Handle: RePEc:spr:joptap:v:128:y:2006:i:1:d:10.1007_s10957-005-7557-y
    DOI: 10.1007/s10957-005-7557-y
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    Citations

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    Cited by:

    1. Gürkan, G. & Ozdemir, O. & Smeers, Y., 2013. "Generation Capacity Investments in Electricity Markets : Perfect Competition," Discussion Paper 2013-045, Tilburg University, Center for Economic Research.
    2. Mak, Davye & Choeum, Daranith & Choi, Dae-Hyun, 2020. "Sensitivity analysis of volt-VAR optimization to data changes in distribution networks with distributed energy resources," Applied Energy, Elsevier, vol. 261(C).
    3. Vasnev, Andrey L., 2010. "Sensitivity of GLS estimators in random effects models," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1252-1262, May.
    4. Enrique Castillo & Roberto Mínguez & Antonio Conejo & Beatriz Pérez & Oscar Fontenla, 2013. "Estimating the parameters of a fatigue model using Benders’ decomposition," Annals of Operations Research, Springer, vol. 210(1), pages 309-331, November.
    5. Gürkan, G. & Ozdemir, O. & Smeers, Y., 2013. "Generation Capacity Investments in Electricity Markets : Perfect Competition," Other publications TiSEM 97828b2c-3630-4b66-8ac9-f, Tilburg University, School of Economics and Management.
    6. Michael K. McWilliam & Antariksh C. Dicholkar & Frederik Zahle & Taeseong Kim, 2022. "Post-Optimum Sensitivity Analysis with Automatically Tuned Numerical Gradients Applied to Swept Wind Turbine Blades," Energies, MDPI, vol. 15(9), pages 1-19, April.
    7. Huang, Chunyi & Zhang, Mingzhi & Wang, Chengmin & Xie, Ning & Yuan, Zhao, 2022. "An interactive two-stage retail electricity market for microgrids with peer-to-peer flexibility trading," Applied Energy, Elsevier, vol. 320(C).
    8. E. Castillo & A. Conejo & C. Castillo & R. Mínguez, 2007. "Closed formulas in local sensitivity analysis for some classes of linear and non-linear problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(2), pages 355-371, December.
    9. Roberto Mínguez & Antonio Conejo & Enrique Castillo, 2013. "Optimal engineering design via Benders’ decomposition," Annals of Operations Research, Springer, vol. 210(1), pages 273-293, November.
    10. Castillo, Enrique & Menéndez, José María & Sánchez-Cambronero, Santos, 2008. "Predicting traffic flow using Bayesian networks," Transportation Research Part B: Methodological, Elsevier, vol. 42(5), pages 482-509, June.
    11. Sankaranarayanan, Sriram & Feijoo, Felipe & Siddiqui, Sauleh, 2018. "Sensitivity and covariance in stochastic complementarity problems with an application to North American natural gas markets," European Journal of Operational Research, Elsevier, vol. 268(1), pages 25-36.
    12. Castillo, Enrique & Mínguez, Roberto & Castillo, Carmen, 2008. "Sensitivity analysis in optimization and reliability problems," Reliability Engineering and System Safety, Elsevier, vol. 93(12), pages 1788-1800.

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