Sums of Separable and Quadratic Polynomials
Published Online:4 Oct 2022https://doi.org/10.1287/moor.2022.1295
References
- [1] (2018) DC decomposition of nonconvex polynomials with algebraic techniques. Math. Programming 169(1):69–94.Crossref, Google Scholar
- [2] (2012) A convex polynomial that is not sos-convex. Math. Programming 135(1):275–292.Crossref, Google Scholar
- [3] (2013) A complete characterization of the gap between convexity and sos-convexity. SIAM J. Optim. 23(2):811–833.Crossref, Google Scholar
- [4] (2013) NP-hardness of deciding convexity of quartic polynomials and related problems. Math. Programming 137(1):453–476.Crossref, Google Scholar
- [5] (2007) Explicit SOS decompositions of univariate polynomial matrices and the Kalman-Yakubovich-Popov lemma. IEEE Conf. Decision Control (IEEE), 5660–5665. https://ieeexplore.ieee.org/document/4434001.Google Scholar
- [6] (2002) A Frank–Wolfe type theorem for convex polynomial programs. Comput. Optim. Appl. 22(1):37–48.Crossref, Google Scholar
- [7] (1999) Nonlinear Programming (Athena Scientific, Belmont, MA).Google Scholar
- [8] (2009) Convex forms that are not sums of squares. Preprint, submitted October 5, https://arxiv.org/abs/0910.0656.Google Scholar
- [9] (2012) Semidefinite Optimization and Convex Algebraic Geometry (SIAM). https://epubs.siam.org/doi/abs/10.1137/1.9781611972290.ch1.Crossref, Google Scholar
- [10] (2004) Convex Optimization (Cambridge University Press). https://www.cambridge.org/gb/academic/subjects/statistics-probability/optimization-or-and-risk/convex-optimization?format=HB.Crossref, Google Scholar
- [11] (1977) Extremal positive semidefinite forms. Mathematische Annalen 231(1):1–18.Crossref, Google Scholar
- [12] (1977) An old question of Hilbert. Queen’s Papers Pure Appl. Math. 46(4):385–405.Google Scholar
- [13] (1980) Real zeros of positive semidefinite forms. I. Mathematische Zeitschrift 171(1):1–26.Crossref, Google Scholar
- [14] (1995) Sums of squares of real polynomials. Proc. Sympos. Pure Math. 58(2):103–126. https://www.ams.org/books/pspum/062.1/.Google Scholar
- [15] (2020) Shape-constrained regression using sum of squares polynomials. Preprint, submitted April 8, https://arxiv.org/abs/2004.03853.Google Scholar
- [16] (2011) On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems. SIAM J. Optim. 21(3):824–832.Crossref, Google Scholar
- [17] (2002) Computers and Intractability, vol. 29 (WH Freeman, New York).Google Scholar
- [18] (2020) Applications of sum of squares polynomials. Sum of Squares: Theory and Applications, vol. 77. https://bookstore.ams.org/psapm-77/.Google Scholar
- [19] (2010) Semidefinite representation of convex sets. Math. Programming 122(1):21–64.Crossref, Google Scholar
- [20] (2005) Detecting global optimality and extracting solutions in GloptiPoly. Positive Polynomials in Control (Springer, Berlin), 293–310.Crossref, Google Scholar
- [21] (1888) Über die Darstellung definiter Formen als Summe von Formenquadraten. Mathematische Annalen 32(3):342–350.Crossref, Google Scholar
- [22] (2001) Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3):796–817.Crossref, Google Scholar
- [23] (2009) Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4):1995–2014.Crossref, Google Scholar
- [24] (2010) Moments, Positive Polynomials and Their Applications, vol. 1 (Imperial College Press). https://www.worldscientific.com/worldscibooks/10.1142/p665#t=aboutBook.Google Scholar
- [25] (2009) Sums of squares, moment matrices and optimization over polynomials. Emerging Applications of Algebraic Geometry(Springer, New York), 157–270.Crossref, Google Scholar
- [26] MOSEK (2019) Mosek optimization toolbox for matlab. User’s Guide and Reference Manual, version 4.Google Scholar
- [27] (1967) The arithmetic-geometric inequality. Proc. Sympos. Inequalities, Wright-Patterson Air Force Base, OH, 1965 (Academic Press, New York), 205–224.Google Scholar
- [28] (1987) Some NP-complete problems in quadratic and nonlinear programming. Math. Programming 39:117–129.Crossref, Google Scholar
- [29] (1995) Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2):227–234.Crossref, Google Scholar
- [30] (2006) Minimizing polynomials via sum of squares over the gradient ideal. Math. Programming 106(3):587–606.Crossref, Google Scholar
- [31] (2000) Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Unpublished PhD thesis, California Institute of Technology, Pasadena.Google Scholar
- [32] (2008) Computing sum of squares decompositions with rational coefficients. Theoretical Comput. Sci. 409(2):269–281.Crossref, Google Scholar
- [33] (2000) Polynomials that are positive on an interval. Trans. Amer. Math. Soc. 352(10):4677–4692.Crossref, Google Scholar
- [34] (2017) Real stability testing. Proc. Innovations Theoretical Comput. Sci. https://drops.dagstuhl.de/opus/volltexte/2017/8202/pdf/LIPIcs-ITCS-2017-0.pdf.Google Scholar
- [35] (1978) Extremal PSD forms with few terms. Duke Math. J. 45(2):363–374.Crossref, Google Scholar
- [36] (2000) Some concrete aspects of Hilbert’s 17th problem. Contemporary Math. 253:251–272.Crossref, Google Scholar
- [37] (1969) Some definite polynomials which are not sums of squares of real polynomials. Notices Amer. Math. Soc. 16:554. https://www.ams.org/journals/proc/1989-106-04/S0002-9939-1989-0976367-8/.Google Scholar
- [38] (2012) Convexity and Optimization in Finite Dimensions I, vol. 163 (Springer Science & Business Media). https://link.springer.com/book/10.1007/978-3-642-46216-0.Google Scholar

