Abstract
We study an extended trust region subproblem minimizing a nonconvex function over the hollow ball \(r \le \Vert x\Vert \le R\) intersected with a full-dimensional second order cone (SOC) constraint of the form \(\Vert x - c\Vert \le b^T x - a\). In particular, we present a class of valid cuts that improve existing semidefinite programming (SDP) relaxations and are separable in polynomial time. We connect our cuts to the literature on the optimal power flow (OPF) problem by demonstrating that previously derived cuts capturing a convex hull important for OPF are actually just special cases of our cuts. In addition, we apply our methodology to derive a new class of closed-form, locally valid, SOC cuts for nonconvex quadratic programs over the mixed polyhedral-conic set \(\{x \ge 0 : \Vert x \Vert \le 1 \}\). Finally, we show computationally on randomly generated instances that our cuts are effective in further closing the gap of the strongest SDP relaxations in the literature, especially in low dimensions.
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Notes
This differs from other papers, which often define RLT constraints only for explicitly given valid linear constraints, of which (1) has none. So, for the sake of generality, we have defined the RLT constraints allowing for implicit valid linear constraints.
Indeed, our initial motivation for this paper was the desire to understand the inequalities in [9] more fully.
We provide Matlab code for these manipulations in the file chenetal/verify_chenetal.m at the website https://github.com/A-Eltved/strengthened_sdr.
Indeed, for any closed, convex cone \(\mathcal{K}\) and dual cone \(\mathcal{K}^*\), given \(\hat{x} \in \text {int}(\mathcal{K})\), we claim the truncation \(\mathcal{K}^* \cap \{ s : \hat{x}^T s \le 1 \}\) is bounded. Specifically, its recession cone \(\mathcal{K}^* \cap \{ s : \hat{x}^T s \le 0 \} = \{0\}\). If not, then some nonzero \(\tilde{s} \in \mathcal{K}^*\) satisfies \(\hat{x}^T \tilde{s} \le 0\). Because \(\hat{x}\) is interior, for sufficiently small \(\epsilon > 0\), the point \(\tilde{x} := \hat{x} - \epsilon \tilde{s}\) satisfies \(\tilde{x} \in \mathcal{K}\) and \(\tilde{x}^T \tilde{s} < 0\). However, this contradicts the fact that \(\tilde{s} \in \mathcal{K}^*\).
We refer the reader to our GitHub site (https://github.com/A-Eltved/strengthened_sdr) for the full random-generation procedure.
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Acknowledgements
The authors acknowledge the support of their respective universities, which allowed the first author to visit the second author in 2019–20, when this research was initiated. In addition, the authors wish to thank Dan Bienstock for constructive comments as well as two anonymous referees, whose suggestions improved the paper significantly.
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Eltved, A., Burer, S. Strengthened SDP relaxation for an extended trust region subproblem with an application to optimal power flow. Math. Program. 197, 281–306 (2023). https://doi.org/10.1007/s10107-021-01737-9
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DOI: https://doi.org/10.1007/s10107-021-01737-9