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.
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Notes
We refer to this condition as SOSC-NSOCP in order to distinguish it from SOSC for NLP.
Notice that \(g_i(x) = \big ( g_{i0}(x), \overline{g_i(x)} \big ) = \big ( g_{i,1}(x),\ldots ,g_{i,m_i}(x) \big )\), i.e., \(g_{i0}(x) = g_{i,1}(x)\). Similarly, we have \(\lambda _i = (\lambda _{i0},\bar{\lambda }_i) = (\lambda _{i,1},\ldots ,\lambda _{i,m_i})\), i.e., \(\lambda _{i0} = \lambda _{i,1}\).
Note the difference between \(\mathcal {K}^{\ell }\) and \(\mathcal {K}_\ell \). The former denotes the second-order cone in \(\mathbb {R}^{\ell }\), and the latter means the \(\ell \)th second-order cone in the Cartesian product \(\mathcal {K}= \mathcal {K}_1 \times \cdots \times \mathcal {K}_r\).
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Acknowledgments
This work was supported by Grant-in-Aid for Young Scientists (B) (26730012) and for Scientific Research (C) (26330029) from Japan Society for the Promotion of Science.
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Communicated by Florian Potra.
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Fukuda, E.H., Fukushima, M. The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming. J Optim Theory Appl 170, 394–418 (2016). https://doi.org/10.1007/s10957-016-0904-3
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DOI: https://doi.org/10.1007/s10957-016-0904-3
Keywords
- Karush–Kuhn–Tucker conditions
- Nonlinear second-order cone programming
- Second-order sufficient condition
- Slack variables