Abstract
The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including global optimization of polynomials and rational functions, option pricing in finance, constructing quadrature schemes for numerical integration, and distributionally robust optimization. A usual solution approach, due to J.B. Lasserre, is to approximate the convex cone of positive Borel measures by finite dimensional outer and inner conic approximations. We will review some results on these approximations, with a special focus on the convergence rate of the hierarchies of upper and lower bounds for the general problem of moments that are obtained from these inner and outer approximations.
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- 1.
We only deal with the GPM in a restricted setting; more general versions of the problem are studied in, e.g., [54].
- 2.
Formally, we consider the usual Borel σ-algebra, say \(\mathcal {B}\), on \(\mathbb {R}^n\), i.e., the smallest (or coarsest) σ-algebra that contains the open sets in \(\mathbb {R}^n\). A positive, finite Borel measure μ is a nonnegative-valued set function on \(\mathcal {B}\), that is countably additive for disjoint sets in \(\mathcal {B}\). The support of μ is the set, denoted Supp(μ), and defined as the smallest closed set S such that \(\mu (\mathbb {R}^n\setminus S) = 0.\)
- 3.
This result is of independent interest in the study of simulated annealing algorithms.
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Acknowledgements
This work has been supported by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement 813211 (POEMA).
The authors would like to thank Fernando Mario de Oliveira Filho for insightful discussions on the duality theory of the GPM.
Note Added in Proof Some of the above mentioned questions have been recently addressed. In particular, the results in Table 2 for the inner approximation bounds have been sharpened. Namely, the convergence rate in O(1∕r 2) has been extended for the sphere in [15] and for the hypercube equipped with more measures in [55]. A sharper rate in \(O(\log ^2r/r^2)\) for convex bodies and in \(O(\log r/r)\) for compact sets with an interior condition is shown in [55]. In addition, the convergence rate O(1∕r 2) is shown in [23] for the outer approximation bounds in the case of the unit sphere.
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Klerk, E.d., Laurent, M. (2019). A Survey of Semidefinite Programming Approaches to the Generalized Problem of Moments and Their Error Analysis. In: Araujo, C., Benkart, G., Praeger, C., Tanbay, B. (eds) World Women in Mathematics 2018. Association for Women in Mathematics Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-030-21170-7_1
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