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Matrix convex hulls of free semialgebraic sets

Authors: J. William Helton, Igor Klep and Scott McCullough
Journal: Trans. Amer. Math. Soc. 368 (2016), 3105-3139
MSC (2010): Primary 46L07, 14P10, 90C22; Secondary 13J30, 46L89
Published electronically: July 10, 2015
MathSciNet review: 3451871
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Abstract: This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $ \mathcal {C}$ and their projections $ \hat {\mathcal {C}}$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, a cornerstone of real algebraic geometry fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the matrix convex hull of free semialgebraic sets.

This article presents the construction of a sequence $ {\mathcal {C}}^{(d)}$ of LMI domains in increasingly many variables whose projections $ \hat {\mathcal {C}}^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $ \mathcal {D}_p=\{X: p(X)\succeq 0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well-known classical (commutative) convex hulls do not produce the convex hulls in the free case. This failure is illustrated here on one of the simplest free nonconvex $ \mathcal {D}_p$.

A basic question is which free sets $ \hat {\mathcal {S}}$ are the projection of a free semialgebraic set $ \mathcal {S}$? Techniques and results of this paper bear upon this question which is open even for free convex sets.

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Additional Information

J. William Helton
Affiliation: Department of Mathematics, University of California, San Diego, California 92093

Igor Klep
Affiliation: Department of Mathematics, The University of Auckland, New Zealand

Scott McCullough
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Keywords: Convex hull, Linear Matrix Inequality (LMI), LMI domain, spectrahedron, spectrahedrop, semialgebraic set, free real algebraic geometry, noncommutative polynomial
Received by editor(s): February 8, 2014
Published electronically: July 10, 2015
Additional Notes: The first author’s research was supported by National Science Foundation (NSF) grant DMS 1201498, and the Ford Motor Co.
The second author was supported by the Marsden Fund Council of the Royal Society of New Zealand and partially supported by Slovenian Research Agency grants P1-0222, L1-4292 and L1-6722. Part of this research was done while the author was on leave from the University of Maribor
The third author’s research was supported by NSF grant DMS 1101137
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