Abstract
Let V be a semialgebraic set parameterized as
for quadratic polynomials f 0,…,f m and a subset T of Rn. This paper studies semidefinite representation of the convex hull conv(V) or its closure, i.e., describing conv(V) by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by a single quadratic constraint, we prove that conv(V) is equal to the first-order moment type semidefinite relaxation of V, up to taking closures. Similar results hold when every f i is a quadratic form and T is defined by two homogeneous (modulo constants) quadratic constraints, or when all f i are quadratic rational functions with a common denominator and T is defined by a single quadratic constraint, under some proper conditions.
Mathematics Subject Classification. 14P10, 90C22, 90C25.
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References
C. Bayer and J. Teichmann. The proof of Tchakaloff’s Theorem. Proc. Amer. Math. Soc., 134(2006), 3035–3040.
L. Fialkow and J. Nie. Positivity of Riesz functionals and solutions of quadratic and quartic moment problems. J. Functional Analysis, Vol. 258, No. 1, pp. 328–356, 2010
S. He, Z. Luo, J. Nie and S. Zhang. Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization. SIAM Journal on Optimization, Vol. 19, No. 2, pp. 503–523, 2008.
J.W. Helton and J. Nie. Semidefinite representation of convex sets. Mathematical Programming, Ser. A, Vol. 122, No. 1, pp. 21–64, 2010.
J.W. Helton and J. Nie. Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM Journal on Optimization, Vol. 20, No. 2, pp. 759–791, 2009.
J.W. Helton and J. Nie. Structured semidefinite representation of some convex sets. Proceedings of 47th IEEE Conference on Decision and Control, pp. 4797–4800, Cancun, Mexico, Dec. 9–11, 2008.
D. Henrion. Semidefinite representation of convex hulls of rational varieties. Acta Applicandae Mathematicae, Vol. 115, No. 3, pp. 319–327, 2011.
J. Lasserre. Convex sets with semidefinite representation. Mathematical Programming, Vol. 120, No. 2, pp. 457–477, 2009.
10] J. Lasserre. Convexity in semi-algebraic geometry and polynomial optimization. SIAM Journal on Optimization, Vol. 19, No. 4, pp. 1995–2014, 2009.
J. Nie. First order conditions for semidefinite representations of convex sets defined by rational or singular polynomials. Mathematical Programming, Ser. A, Vol. 131, No. 1, pp. 1–36, 2012.
J. Nie. Polynomial matrix inequality and semidefinite representation. Mathematics of Operations Research, Vol. 36, No. 3, pp. 398–415, 2011.
P. Parrilo. Exact semidefinite representation for genus zero curves. Talk at the Banff workshop "Positive Polynomials and Optimization", Banff, Canada, October 8–12, 2006.
G. Pataki. On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Mathematics of Operations Research, 23 (2), 339–358, 1998.
M. Putinar. Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42 (1993) 203–206.
K. Schmüdgen. The K-moment problem for compact semialgebraic sets. Math. Ann. 289 (1991), 203–206.
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Dedicated to Bill Helton on the occasion of his 65th birthday.
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Nie, J. (2012). Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_18
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DOI: https://doi.org/10.1007/978-3-0348-0411-0_18
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