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The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming

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  • Ellen H. Fukuda

    (Kyoto University)

  • Masao Fukushima

    (Nanzan University)

Abstract

In traditional nonlinear programming, the technique of converting a problem with inequality constraints into a problem containing only equality constraints, by the addition of squared slack variables, is well known. Unfortunately, it is considered to be an avoided technique in the optimization community, since the advantages usually do not compensate for the disadvantages, like the increase in the dimension of the problem, the numerical instabilities, and the singularities. However, in the context of nonlinear second-order cone programming, the situation changes, because the reformulated problem with squared slack variables has no longer conic constraints. This fact allows us to solve the problem by using a general-purpose nonlinear programming solver. The objective of this work is to establish the relation between Karush–Kuhn–Tucker points of the original and the reformulated problems by means of the second-order sufficient conditions and regularity conditions. We also present some preliminary numerical experiments.

Suggested Citation

  • Ellen H. Fukuda & Masao Fukushima, 2016. "The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 394-418, August.
    • Handle: RePEc:spr:joptap:v:170:y:2016:i:2:d:10.1007_s10957-016-0904-3
    DOI: 10.1007/s10957-016-0904-3
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    References listed on IDEAS

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    1. Robert Fourer & David M. Gay & Brian W. Kernighan, 1990. "A Modeling Language for Mathematical Programming," Management Science, INFORMS, vol. 36(5), pages 519-554, May.
    2. Y. J. Liu & L. W. Zhang, 2008. "Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones," Journal of Optimization Theory and Applications, Springer, vol. 139(3), pages 557-575, December.
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    Cited by:

    1. Bruno F. Lourenço & Ellen H. Fukuda & Masao Fukushima, 2018. "Optimality Conditions for Problems over Symmetric Cones and a Simple Augmented Lagrangian Method," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1233-1251, November.
    2. Katsuhiro Tanaka, 2019. "Forecasting plausible scenarios and losses in interest rate targeting using mathematical optimization," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(03), pages 1-24, September.

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