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In SDP relaxations, inaccurate solvers do robust optimization

Abstract : We interpret some wrong results (due to numerical inaccuracies) already observed when solving SDP-relaxations for polynomial optimization on a double precision floating point SDP solver. It turns out that this behavior can be explained and justified satisfactorily by a relatively simple paradigm. In such a situation, the SDP solver (and not the user) performs some `robust optimization' without being told to do so. Instead of solving the original optimization problem with nominal criterion $f$, it uses a new criterion $\tilde{f}$ which belongs to a ball $\mathbf{B}_\infty(f,\varepsilon)$ of small radius $\varepsilon>0$, centered at the nominal criterion $f$ in the parameter space. In other words the resulting procedure can be viewed as a `$\max-\min$' robust optimization problem with two players (the solver which maximizes on $\mathbf{B}_\infty(f,\varepsilon)$ and the user who minimizes over the original decision variables). A mathematical rationale behind this `autonomous' behavior is described.
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https://hal.archives-ouvertes.fr/hal-01915976
Contributor : Victor Magron <>
Submitted on : Thursday, November 8, 2018 - 10:33:01 AM
Last modification on : Wednesday, April 8, 2020 - 3:46:53 PM

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Jean-Bernard Lasserre, Victor Magron. In SDP relaxations, inaccurate solvers do robust optimization. SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2019, 29 (3), pp.2128-2145. ⟨10.1137/18M1225677⟩. ⟨hal-01915976⟩

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