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Convergence rates for the moment-SoS hierarchy (2402.00436v3)

Published 1 Feb 2024 in math.OC

Abstract: We introduce a comprehensive framework for analyzing convergence rates for infinite dimensional linear programming problems (LPs) within the context of the moment-sum-of-squares hierarchy. Our primary focus is on extending the existing convergence rate analysis, initially developed for static polynomial optimization, to the more general and challenging domain of the generalized moment problem. We establish an easy-to-follow procedure for obtaining convergence rates. Our methodology is based on, firstly, a state-of-the-art degree bound for Putinar's Positivstellensatz, secondly, quantitative polynomial approximation bounds, and, thirdly, a geometric Slater condition on the infinite dimensional LP. We address a broad problem formulation that encompasses various applications, such as optimal control, volume computation, and exit location of stochastic processes. We illustrate the procedure at these three problems and, using a recent improvement on effective versions of Putinar's Positivstellensatz, we improve existing convergence rates.

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References (51)
  1. Multivariate simultaneous approximation. Constructive approximation, 18(4):569–577, 2002.
  2. On the effective Putinar’s Positivstellensatz and moment approximation. Mathematical Programming, 200(1):71–103, 2023.
  3. On łojasiewicz inequalities and the effective Putinar’s Positivstellensatz. arXiv: 2212.09551 [math.AC], 2022.
  4. Degree bounds for Putinar’s Positivstellensatz on the hypercube. SIAM Journal on Applied Algebra and Geometry, 8(1):1–25, 2024.
  5. Alexander Barvinok. A Course in Convexity, volume 54 of Graduate Studies in Mathematics. American Mathematical Society, 2002.
  6. Eman Samir Bhaya. Interpolating operators for multiapproximation. Journal of Mathematics and Statistics, 6(3):240–245, 2010.
  7. Gibbs phenomena for some classical orthogonal polynomials. Journal of Mathematical Analysis and Applications, 505(1):125574, 2022.
  8. Exact solutions to super resolution on semi-algebraic domains in higher dimensions. IEEE Transactions on Information Theory, 63:621–630, 2017.
  9. Lawrence C Evans. An introduction to stochastic differential equations, volume 82. American Mathematical Soc., 2012.
  10. The Sum-of-Squares hierarchy on the sphere and applications in quantum information theory. Mathematical Programming, 190:331–360, 2021.
  11. Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization. SIAM Journal on Applied Dynamical Systems, 15(4):1962–1988, 2016.
  12. Elliptic partial differential equations of second order, volume 224. Springer, 1977.
  13. David Goluskin. Bounding extrema over global attractors using polynomial optimisation. Nonlinearity, 33(9):4878, 2020.
  14. Moment-SOS hierarchy and exit location of stochastic processes. HAL preprint: hal-03110452, 2023.
  15. Occupation measure relaxations in variational problems: the role of convexity. arXiv preprint arXiv:2303.02434, 2023.
  16. Approximate volume and integration for basic semialgebraic sets. SIAM Review, 51:722–743, 2009.
  17. Converse Lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems. In 2021 60th IEEE Conference on Decision and Control (CDC), pages 5312–5319. IEEE, 2021.
  18. Milan Korda. Computing controlled invariant sets from data using convex optimization. SIAM Journal on Control and Optimization, 58(5):2871–2899, 2020.
  19. Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets. Optimization Letters, 12(3):435–442, 2018.
  20. Inner approximations of the region of attraction for polynomial dynamical systems. IFAC Proceedings Volumes, 46(23):534–539, 2013.
  21. Convex computation of the maximum controlled invariant set for polynomial control systems. SIAM Journal on Control and Optimization, 52(5):2944–2969, 2014.
  22. Convergence rates of moment-sum-of-squares hierarchies for optimal control problems. Systems & Control Letters, 100:1–5, 2017.
  23. Convex computation of extremal invariant measures of nonlinear dynamical systems and Markov processes. Journal of Nonlinear Science, 31(1):1–26, 2021.
  24. Urysohn in action: separating semialgebraic sets by polynomials. arXiv preprint arXiv:2207.00570, 2022.
  25. Convergence rates for sums-of-squares hierarchies with correlative sparsity. arXiv preprint arXiv:2303.14824, 2023.
  26. Jean B. Lasserre. An explicit exact SDP relaxation for nonlinear 0-1 programs. In Integer Programming and Combinatorial Optimization conference, pages 293–303, 2001.
  27. Jean B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11:796–817, 2001.
  28. Jean B. Lasserre. Moments, positive polynomials and their applications, volume 1. World Scientific, 2009.
  29. Jean B. Lasserre. An introduction to polynomial and semi-algebraic optimization, volume 52. Cambridge University Press, 2015.
  30. Jean B Lasserre. Tractable approximations of sets defined with quantifiers. Mathematical Programming, 151:507–527, 2015.
  31. Jean B. Lasserre. Computing Gaussian and exponential measures of semi-algebraic sets. Advances in Applied Mathematics, 91:137–163, 2017.
  32. Nonlinear optimal control via occupation measures and LMI-relaxations. SIAM journal on control and optimization, 47(4):1643–1666, 2008.
  33. John M. Lee. Introduction to smooth manifolds. Graduate Texts in Mathematics. Springer, second edition, 2013.
  34. Semidefinite approximations of projections and polynomial images of semialgebraic sets. SIAM Journal on Optimization, 25(4):2143–2164, 2015.
  35. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control and Related Fields, 10(1):113–140, 2020.
  36. Unsafe probabilities and risk contours for stochastic processes using convex optimization. arXiv preprint arXiv:2401.00815, 2023.
  37. Peak Value-at-Risk estomation for stochastic differential equations using occupation measures. In IEEE Conference on Decision and Control, 2023.
  38. Jiawang Nie. Optimality conditions and finite convergence in Lasserre’s hierarchy. Mathematical Programming A, 146:97–121, 2014.
  39. On the complexity of Putinar’s Positivstellensatz. Journal of Complexity, 23(1):135–150, 2007.
  40. Inner approximations of the maximal positively invariant set for polynomial dynamical systems. IEEE Control Systems Letters, 3(3):733–738, 2019.
  41. Mihai Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42(3):969–984, 1993.
  42. Exploiting symmetries in SDP-relaxations for polynomial optimization. Mathematics of Operations Research, 38(1):122–141, 2013.
  43. Sparse moment-sum-of-squares relaxations for nonlinear dynamical systems with guaranteed convergence. arXiv preprint arXiv:2012.05572, 2020.
  44. Converging outer approximations to global attractors using semidefinite programming. Automatica, 134:109900, 2021.
  45. Morton L. Slater. Lagrange multipliers revisited. Technical report, Cowles Commission, 1950. Discussion Paper No. 403.
  46. Lucas Slot. Sum-of-Squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel. arXiv preprint arXiv: 2111.04610, 2021.
  47. Matteo Tacchi. Moment-SoS hierarchy for large scale set approximation. Application to power systems transient stability analysis. PhD thesis, Toulouse University, 2021.
  48. Matteo Tacchi. Convergence of Lasserre’s hierarchy: the general case. Optimization Letters, 16:1015–1033, 2022.
  49. Stokes, Gibbs and volume computation of semi-algebraic sets. Discrete and Computational Geometry, 69:260–283, 2023.
  50. Exploiting sparsity for semi-algebraic set volume computation. Foundations of Computational Mathematics, pages 1–49, 2022.
  51. John Urbas. Lecture on second order linear partial differential equations. In Instructional Workshop on Analysis and Geometry, Part 1, volume 34, pages 39–76. Australian National University, Mathematical Sciences Institute, 1996.
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