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On an extension of Pólya’s Positivstellensatz. (English) Zbl 1379.11037
Summary: In this paper, we provide a generalization of a Positivstellensatz by G. Pólya [Vierteljahrsschrift Zürich 73, 141–145 (1928; JFM 54.0138.01)]. We show that if a homogeneous polynomial is positive over the intersection of the non-negative orthant and a given basic semialgebraic cone (excluding the origin), then there exists a “Pólya type” certificate for non-negativity. The proof of this result uses the original Positivstellensatz by Pólya, and a Positivstellensatz by M. Putinar and F.-H. Vasilescu [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 7, 585–589 (1999; Zbl 0973.14031)].

11E25 Sums of squares and representations by other particular quadratic forms
14P05 Real algebraic sets
14P10 Semialgebraic sets and related spaces
90C30 Nonlinear programming
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[1] Bomze, IM, Copositive optimization-recent developments and applications, Eur. J. Oper. Res., 216, 509-520, (2012) · Zbl 1262.90129
[2] Cplex optimizer. http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/. Apr 2014 · Zbl 1035.90058
[3] Dickinson, P.J.C.: The copositive cone, the completely positive cone and their generalisations. PhD thesis, University of Groningen, Groningen, The Netherlands (2013) · Zbl 1161.68480
[4] Klerk, E; Laurent, M; Parrilo, PA, A PTAS for the minimization of polynomials of fixed degree over the simplex, Theor. Comput. Sci., 361, 210-225, (2006) · Zbl 1115.90042
[5] Klerk, E; Pasechnik, DV, Approximation of the stability number of a graph via copositive programming, SIAM J. Optim., 12, 875-892, (2002) · Zbl 1035.90058
[6] Dong, H, Symmetric tensor approximation hierarchies for the completely positive cone, SIAM J. Optim., 23, 1850-1866, (2013) · Zbl 1291.90129
[7] Dickinson, P.J.C., Povh, J.: New linear and positive semidefinite programming based approximation hierarchies for polynomial optimisation. Preprint, submitted. Available at http://www.optimization-online.org/DB_HTML/2013/06/3925.html (2013)
[8] Dür, M; Diehl, M (ed.); Glineur, F (ed.); Jarlebring, E (ed.); Michiels, W (ed.), Copositive programming-a survey, 3-20, (2010), Berlin
[9] Faybusovich, L.: Global optimization of homogeneous polynomials on the simplex and on the sphere. Frontiers in Global Optimization, In: Floudas, C.A., Pardalos, P. (eds.) Nonconvex Optimization and its Application, vol. 74, pp. 109-121. Kluwer, Boston (2004) · Zbl 1165.90592
[10] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1988) · Zbl 0634.26008
[11] Krivine, J-L, Anneaux préordonnés, J. Anal. Math., 12, 307-326, (1964) · Zbl 0134.03902
[12] Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. Emerging Applications of Algebraic Geometry, In: Putinar, M., Sullivant, S. (eds.) IMA Volumes in Mathematics and its Applications, vol. 149, pp. 157-270. Springer, New York (2009) · Zbl 1163.13021
[13] Decision tree for optimization software. http://plato.asu.edu/guide.html. Apr 2014 · Zbl 1262.90129
[14] Murray, M; Tim, N, Positivstellensätze for real function algebras, Math. Z., 270, 889-901, (2012) · Zbl 1242.13029
[15] Mosek optimization software. http://mosek.com/. Apr 2014 · Zbl 0796.12002
[16] Nie, J; Schweighofer, M, On the complexity of putinar’s positivstellensatz, J. Complex., 23, 135-150, (2007) · Zbl 1143.13028
[17] Parrilo, P.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology (2000)
[18] Pólya, G.: Über positive darstellung von polynomen vierteljschr. In: Naturforsch. Ges. Zürich, 73: 141-145, 1928. In: Boas, R.P. (ed.) Collected Papers. vol. 2, pp. 309-313. MIT Press, Cambridge. Available at http://hal.archives-ouvertes.fr/docs/00/60/96/87/PDF/RAG2011-Rennes.pdf (1974) · Zbl 1291.90129
[19] Povh, J.: Towards the Optimum by Semidefinite and Copositive Programming: New Approach to Approximate Hard Optimization Problems. VDM, Saarbrücken (2009)
[20] Powers, V.: Positive polynomials and sums of squares: theory and practice. Real Algebraic, Geometry, p. 77 (2011) · Zbl 1115.90042
[21] Powers, V., Reznick, B.: A new bound for Pólya’s theorem with applications to polynomials positive on polyhedra. J. Pure Appl. Algebra, 164(1-2):221-229, (2001) Effective methods in algebraic geometry (Bath, 2000) · Zbl 1075.14523
[22] Putinar, M, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., 42, 969-984, (1993) · Zbl 0796.12002
[23] Putinar, M; Vasilescu, F-H, Positive polynomials on semi-algebraic sets, C. R. Acad. Sci. Ser. I Math., 328, 585-589, (1999) · Zbl 0973.14031
[24] Putinar, M; Vasilescu, F-H, Solving moment problems by dimensional extension, Ann. of Math. (2), 149, 1087-1107, (1999) · Zbl 0939.44003
[25] Reznick, B, Uniform denominators in hilbert’s seventeenth problem, Math. Z., 220, 75-97, (1995) · Zbl 0828.12002
[26] Schmüdgen, K, The \(K\)-moment problem for compact semi-algebraic sets, Math. Ann., 289, 203-206, (1991) · Zbl 0744.44008
[27] Schweighofer, M, On the complexity of schmüdgen’s positivstellensatz, J. Complex., 20, 529-543, (2004) · Zbl 1161.68480
[28] Scheiderer, C.: Positivity and sums of squares: a guide to recent results. Emerging applications of algebraic geometry, volume 149 of IMA Volumes in Mathematics and its Applications, pp. 271-324. Springer, New York (2009) · Zbl 1156.14328
[29] Stengle, G, A nullstellensatz and a positivstellensatz in semialgebraic geometry, Math. Ann., 207, 87-97, (1974) · Zbl 0253.14001
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