Heß, Roxana; Henrion, Didier; Lasserre, Jean-Bernard; Phạm, Tien Son Semidefinite approximations of the polynomial abscissa. (English) Zbl 1351.90127 SIAM J. Control Optim. 54, No. 3, 1633-1656 (2016). Summary: Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is Hölder continuous, and not locally Lipschitz in general, which is a source of numerical difficulties for designing and optimizing control laws. In this paper we propose simple approximations of the abscissa given by polynomials of fixed degree, and hence controlled complexity. Our approximations are computed by a hierarchy of finite-dimensional convex semidefinite programming problems. When their degree tends to infinity, the polynomial approximations converge in \(L^1\) norm to the abscissa, either from above or from below. Cited in 2 Documents MSC: 90C22 Semidefinite programming 90C26 Nonconvex programming, global optimization 26C10 Real polynomials: location of zeros 41A10 Approximation by polynomials Keywords:linear systems control; nonconvex nonsmooth optimization; polynomial approximations; semialgebraic optimization; semidefinite programming Software:Mosek; YALMIP; Polynomial Toolbox PDFBibTeX XMLCite \textit{R. Heß} et al., SIAM J. Control Optim. 54, No. 3, 1633--1656 (2016; Zbl 1351.90127) Full Text: DOI arXiv References: [1] R. B. Ash, {\it Probability and Measure Theory}, 2nd ed., Academic Press, San Diego, 2000. · Zbl 0944.60004 [2] A. Barvinok, {\it A Course in Convexity}, AMS, Providence, RI, 2002. · Zbl 1014.52001 [3] J. Bochnak, M. Coste, and M.-F. Roy, {\it Real Algebraic Geometry}, Springer, Berlin, 1998. · Zbl 0912.14023 [4] J. V. Burke, D. Henrion, A. S. Lewis, and M. L. Overton, {\it Stabilization via nonsmooth, nonconvex optimization}, IEEE Trans. Automat. Control, 51 (2006), pp. 1760-1769. · Zbl 1366.93490 [5] J. V. Burke, A. S. Lewis, and M. L. Overton, {\it Variational analysis of the abscissa mapping for polynomials via the Gauss-Lucas theorem}, J. Global Optim., 28 (2004), pp. 259-268. · Zbl 1134.49309 [6] J. A. Cross, {\it Spectral Abscissa Optimization Using Polynomial Stability Conditions}, Ph.D. thesis, University of Washington, Seattle, 2010. [7] D. Henrion and J. B. Lasserre, {\it Inner approximations for polynomial matrix inequalities and robust stability regions}, IEEE Trans. Automat. Control, 57 (2012), pp. 1456-1467. · Zbl 1369.93460 [8] D. Henrion, D. Peaucelle, D. Arzelier, and M. Šebek, {\it Ellipsoidal approximation of the stability domain of a polynomial}, IEEE Trans. Automat. Control, 48 (2003), pp. 2255-2259. · Zbl 1364.93568 [9] J. B. Lasserre, {\it Moments, Positive Polynomials and Their Applications}, Imperial College Press, London, 2010. · Zbl 1211.90007 [10] M. Laurent, {\it Sums of squares, moment matrices and polynomial optimization}, in Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivan, eds., IMA Vol. Math. Appl., 149, Springer, Berlin, 2009. [11] MOSEK ApS, Copenhagen, Denmark, www.mosek.com [12] J. Löfberg, {\it YALMIP: A Toolbox for Modeling and Optimization in MATLAB}, presented at IEEE CACSD Conference, Taipei, Taiwan, 2004; available online from users.isy.liu.se/johanl/yalmip/. [13] V. A. Zorich, {\it Mathematical Analysis} II, Springer, Berlin, 2004. · Zbl 1071.00003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.