Abstract
In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We characterize the notions of so-called invariancy set and nonlinearity interval, which serve as stability regions of the optimal partition. We then propose, under the strict complementarity condition, an iterative procedure to compute a nonlinearity interval of the optimal partition. Furthermore, under primal and dual nondegeneracy conditions, we show that a boundary point of a nonlinearity interval can be numerically identified from a nonlinear reformulation of the parametric second-order conic optimization problem. Our theoretical results are supported by numerical experiments.
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Notes
In this context, \(\infty \) simply means that the primal problem is infeasible.
In this paper, strong duality means that the duality gap is zero at optimality, and the optimal sets \({\mathcal {P}}^*(\epsilon )\) and \({\mathcal {D}}^*(\epsilon )\) are nonempty.
Here, local boundedness is equivalent to uniform compactness in [23].
See [28, Example 3.1] for an instance of a parametric SDO problem with infinitely many singleton invariancy sets.
For this problem, both the primal and dual optimal set mappings are continuous at the transition point \(\epsilon = 0\).
For the ease of exposition, we simply rule out the existence of an invariancy interval here. Otherwise, as indicated in Remark 5, this procedure can also be applied to invariancy intervals.
Since \({\mathcal {P}}^*(\epsilon ) \times {\mathcal {D}}^*(\epsilon )\) is single-valued and locally bounded at \({\bar{\epsilon }}\), see (8), for every ball \({\mathbb {B}}_r\) of radius r centered at \(\big (x^*({\bar{\epsilon }});y^*({\bar{\epsilon }});s^*({\bar{\epsilon }})\big )\) there exists a neighborhood V of \({\bar{\epsilon }}\) such that \({\mathcal {P}}^*(\epsilon ) \times {\mathcal {D}}^*(\epsilon ) \subseteq {\mathbb {B}}_r\) for every \(\epsilon \in V\), see [34, Proposition 5.12(a)]. Therefore, it is always possible to find such an \({\hat{\epsilon }}\) with both desired properties.
The perturbation theory of invariant subspaces states that the eigenspace associated with the cluster of positive eigenvalues of \(L\big (x^{*i}({\bar{\epsilon }})\big )\) stays near that of \(L\big (x^{*i}({\hat{\epsilon }})\big )\).
On an open set \(U \subseteq {\mathbb {R}}\), a mapping f(x) is real analytic if for any given \(x_0 \in U\)
$$\begin{aligned} f(x)=\sum _{k=0}^{\infty } \frac{\big (f(x_0)\big )^{(k)}}{k!} (x-x_0)^k \end{aligned}$$for all x in a neighborhood of \(x_0\), where \((.)^{(k)}\) denotes the \(k{\mathrm {th}}\)-order derivative. See [25, Definition 1.1.5] for further properties of an analytic mapping.
In fact, using (20), (25), and (29) we can generate an optimal solution \(\big (x^*(\epsilon );y^*(\epsilon );s^*(\epsilon )\big )\) for \((\mathrm {P}_{\epsilon })\) and \((\mathrm {D}_{\epsilon })\), see [26, Section 3], which then proves to be unique for every \(\epsilon \) sufficiently close to \({\bar{\epsilon }}\).
Interestingly, the duality gap is zero at the boundary point \(\epsilon = 1-\sqrt{2}\), with the finite optimal value \(1/\sqrt{2}\). However, the optimal value of (32) is not attained, and its dual fails to have a strictly feasible solution.
As defined in [28, Definition 1.1], \(\big (X^*(\epsilon ),y^*(\epsilon ),S^*(\epsilon )\big )\) is a maximally complementary optimal solution of \((\mathrm {P}'_{\epsilon })\) and \((\mathrm {D}'_{\epsilon })\) if \({{\,\mathrm{rank}\,}}\!\big (X^*(\epsilon )\big ) + {{\,\mathrm{rank}\,}}\!\big (S^*(\epsilon )\big )\) is maximal on \(\mathcal {P}'^*(\epsilon ) \times \mathcal {D}'^*(\epsilon )\).
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Acknowledgements
We would like to express our gratitude to the anonymous referees for their insightful comments and suggestions. We are indebted to Professor Saugata Basu for bringing the semi-algebraic properties of transition points to our attention.
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This paper is dedicated to Marco Lopez on the occasion of his 70th birthday.
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This work is supported by Air force Office of Scientific Research (AFOSR) Grant # FA9550-15-1-0222.
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Mohammad-Nezhad, A., Terlaky, T. On the sensitivity of the optimal partition for parametric second-order conic optimization. Math. Program. 189, 491–525 (2021). https://doi.org/10.1007/s10107-021-01690-7
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DOI: https://doi.org/10.1007/s10107-021-01690-7