Abstract
Semidefinite programming (SDP) (Ben-Tal and Nemirovski: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization, SIAM, Philadelphia (2001); Nesterov and Nemirovskii: Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1994); Nemirovskii, A.: Advances in convex optimization: conic programming. Plenary Lecture at the International Congress of Mathematicians (ICM), Madrid, Spain, 22-30 August 2006; Wolkowicz et al.: Handbook of semidefinite programming. Kluwer’s, Boston, MA (2000)) is one of the main advances in convex optimization theory and applications. It has a profound effect on combinatorial optimization, control theory and nonconvex optimization as well as many other disciplines. There are effective numerical algorithms for solving problems presented in terms of Linear Matrix Inequalities (LMIs). A fundamental problem in semidefinite programming concerns the range of its applicability. What convex sets S can be represented by using LMI? This question has several natural variants; the main one is to represent S as the projection of a higher dimensional set that is representable by LMI; such S is called SDP representable (SDr). Here we shall describe these variants and what is known about them. The chapter is a survey of the existing results of (Helton and Nie: Mathematical Programming 122(1):21–64 (2010); Helton and Nie: SIAM Journal on Optimization 20(2):759–791 (2009); Lasserre: Mathematical Programming 120:457–477 (2009)) and related work.
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- 1.
One can replace here “almost every line” by “every line” if one counts multiplicities into the number of intersections, and also counts the intersections at infinity, i.e., replaces the affine real algebraic hypersurface p(x) = 0 in \({\mathbb{R}}^{n}\) by its projective closure in \({\mathbb{P}}^{n}(\mathbb{R})\).
- 2.
The result does not require a basic semialgebraic convex set, a semialgebraic convex set will do, although description of the boundary varieties is a bit more complicated. See Theorem 4.8 and subsequent remarks.
- 3.
In general, f being strictly convex does not imply its Hessian is positive definite.
References
Ahmadi, A.A., Parrilo, P.A.: A convex polynomial that is not sos-convex. To appear in Mathematical Programming
Blekherman, G.: Convex forms that are not sums of squares. Preprint (2009). http://arxiv.org/abs/0910.0656
Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry, Springer (1998)
Branden, P.: Obstructions to determinantal representability. Advances in Mathematics 226, 1202–1212 (2011)
Conway, J.B.: A course in Functional Analysis, Springer Graduate Texts in Math Series (1985)
Ghomi, M.: Optimal Smoothing for Convex Polytopes. Bull. London Math. Soc. 36, 483–492 (2004)
Helton, J.W., Nie, J.: Semidefinite representation of convex sets. Mathematical Programming 122(1), 21–64 (2010)
Helton, J.W., Nie, J.: Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets. SIAM Journal on Optimization 20(2), 759–791 (2009)
Helton, J.W., Nie, J.: Structured Semidefinite Representation of Some Convex Sets. Proceedings of 47th IEEE Conference on Decision and Control (CDC), pp. 4797–4800, Cancun, Mexico, 9-11 December 2008
Helton, J.W., McCullough, S., Vinnikov, V.: Noncommutative convexity arises from linear matrix inequalities. J. Funct. Anal. 240(1), 105–191 (2006)
Helton, W., Vinnikov, V.: Linear matrix inequality representation of sets. Comm. Pure Appl. Math 60(5), 654–674 (2007)
Henrion, D.: Semidefinite representation of convex hulls of rational varieties. LAAS-CNRS Research Report No. 09001, January 2009
Lasserre, J.: Convex sets with semidefinite representation. Mathematical Programming 120, 457–477 (2009)
Lasserre, J.: Convexity in SemiAlgebraic Geometry and Polynomial Optimization. SIAM J. Optim. 19, 1995–2014 (2009)
Lax, P.: Differential equations, difference equations and matrix theory. Communications on Pure and Applied Mathematics 6, 175–194 (1958)
Lewis, A., Parrilo P., Ramana, M.: The Lax conjecture is true. Proc. Amer. Math. Soc. 133(9), 2495–2499 (2005)
Nesterov, Y., Nemirovskii, A.: Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1994)
Nemirovski, A.: Advances in convex optimization: conic programming. Plenary Lecture at the International Congress of Mathematicians (ICM), Madrid, Spain, 22-30 August 2006
Netzer, T., Plaumann, D., Schweighofer, M.: Exposed faces of semidefinitely representable sets. SIAM Journal on Optimization 20(4), 1944–1955 (2010)
Nie, J.: First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials. To appear in Mathematical Programming
Parrilo, P.: Exact semidefinite representation for genus zero curves. Paper presented at the Positive Polynomials and Optimization Workshop, Banff, Canada, 7–12 October 2006
Parrilo, P., Sturmfels, B.: Minimizing polynomial functions. In: Basu, S., Gonzalez-Vega, L. (eds.) Proceedings of the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, March 2001. pp. 83–100. American Mathematical Society (2003)
Putinar, M.: Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42 203–206 (1993)
Ranestad, K. Sturmfels, B.: On the convex hull of a space curve. Advances in Geometry, To appear in Advances in Geometry
Ranestad, K. Sturmfels, B.: The convex hull of a variety. In: Notions of Positivity and the Geometry of Polynomials Branden, P., Passare, M., Putinar, M. (eds.), Trends in Mathematics, Springer Verlag, Basel, pp. 331–344, (2011)
Reznick, B.: Some concrete aspects of Hilbert’s 17{ th} problem. In: Contemp. Math., vol. 253, pp. 251–272. American Mathematical Society (2000)
Sanyal, R., Sottile, F., Sturmfels, B.: Orbitopes. Mathematika 57, 275–314 (2011)
Scheiderer, C.: Convex hulls of curves of genus one. arXiv:1003.4605v1 [math.AG]
Spivak, M.: A comprehensive introduction to differential geometry, vol. II, second edition, Publish or Perish, Inc., Wilmington, Del. (1979)
Acknowledgements
J. William Helton was partially supported by NSF grants DMS-0757212, DMS-0700758 and the Ford Motor Company. Jiawang Nie was partially supported by NSF grants DMS-0757212, DMS-0844775 and Hellman Foundation Fellowship.
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Helton, J.W., Nie, J. (2012). Semidefinite Representation of Convex Sets and Convex Hulls. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_4
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