Noncommutative partially convex rational functions. (English) Zbl 1497.46082
Summary: Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs – \(xy\)-convexity – is also considered. A characterization of \(xy\)-convex polynomials is given.
MSC:
| 46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |
| 46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
| 26B25 | Convexity of real functions of several variables, generalizations |
| 47A63 | Linear operator inequalities |
| 52A41 | Convex functions and convex programs in convex geometry |
| 90C25 | Convex programming |
Keywords:
partial convexity; biconvexity; bilinear matrix inequality (BMI); noncommutative rational function; noncommutative polynomial; realization theoryReferences:
| [1] | Ando, T.: Majorization, doubly stochastic matrices, and comparison of eigenvalues. Linear Algebra Appl. 118 (1989), 163-248. · Zbl 0673.15011 |
| [2] | Ando, T.: Majorizations and inequalities in matrix theory. Linear Algebra Appl. 199 (1994), 17-67. · Zbl 0798.15024 |
| [3] | Balasubramanian, S. and McCullough, S.: Quasi-convex free polynomials. Proc. Amer. Math. Soc. 142 (2014), no. 8, 2581-2591. · Zbl 1391.16028 |
| [4] | Ball, J. A., Groenewald, G. and Malakorn, T.: Structured noncommutative multidimensional linear systems. SIAM J. Control Optim. 44 (2005), no. 4, 1474-1528. · Zbl 1139.93006 |
| [5] | Ball, J. A., Marx, G. and Vinnikov, V.: Noncommutative reproducing kernel Hilbert spaces. J. Funct. Anal. 271 (2016), no. 7, 1844-1920. · Zbl 1350.47014 |
| [6] | Bergman, G.: Rational relations and rational identities in division ring I, II. J. Algebra 43 (1976), no. 1, 252-266 and 267-297. · Zbl 0307.16012 |
| [7] | Bhatia, R.: Matrix analysis. Graduate Texts in Mathematics 169, Springer-Verlag, New York, 1997. |
| [8] | Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V.: Linear matrix inequalities in sys-tem and control theory. SIAM Studies in Applied Mathematics 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. · Zbl 0816.93004 |
| [9] | Brešar, M. and Klep, I.: A local-global principle for linear dependence of noncommutative polynomials. Israel J. Math. 193 (2013), no. 1, 71-82. · Zbl 1293.16021 |
| [10] | Camino, J. F., Helton, J. W., Skelton, R. E. and Ye, J.: Matrix inequalities: a symbolic procedure to determine convexity automatically. Integral Equations Operator Theory 46 (2003), no. 4, 399-454. · Zbl 1046.68139 |
| [11] | Davidson, K. R., Dor-On, A., Shalit, O. M. and Solel, B.: Dilations, inclusions of matrix convex sets, and completely positive maps. Int. Math. Res. Not. IMRN (2017), no. 13, 4069-4130. · Zbl 1405.15024 |
| [12] | Davidson, K. and Kennedy, M.: Noncommutative Choquet theory. Preprint 2019, arXiv: 1905.08436. |
| [13] | de Oliveira, M. C., Helton, J. W., McCullough, S. A. and Putinar, M.: Engineering systems and free semi-algebraic geometry. In Emerging applications of algebraic geometry, 17-61. IMA Vol. Math. Appl. 149, Springer, New York, 2009, · Zbl 1156.14331 |
| [14] | Dym, H., Helton, J. W. and McCullough, S.: Non-commutative polynomials with convex level slices. Indiana Univ. Math. J. 66 (2017), no. 6, 2071-2135. · Zbl 1402.15011 |
| [15] | Ebadian, A., Nikoufar, I. and Eshaghi Gordji, M.: Perspectives of matrix convex functions. Proc. Natl. Acad. Sci. USA 108 (2011), no. 18, 7313-7314. · Zbl 1256.81020 |
| [16] | Effros, E. G.: A matrix convexity approach to some celebrated quantum inequalities. Proc. Natl. Acad. Sci. USA 106 (2009), no. 4, 1006-1008. · Zbl 1202.81018 |
| [17] | Evert, E.: Matrix convex sets without absolute extreme points. Linear Algebra Appl. 537 (2018), 287-301. · Zbl 1386.46044 |
| [18] | Evert, E. and Helton, J. W.: Arveson extreme points span free spectrahedra. Math. Ann. 375 (2019), no. 1-2, 629-653. · Zbl 1440.47056 |
| [19] | Goh, K.-C., Safonov, M. G. and Ly, J. H.: Robust synthesis via bilinear matrix inequalities. Linear matrix inequalities in control theory and applications. Internat. J. Robust Nonlinear Control 6 (1996), no. 9-10, 1079-1095. · Zbl 0861.93015 |
| [20] | Guionnet, A. and Shlyakhtenko, D.: Free diffusions and matrix models with strictly convex interaction. Geom. Funct. Anal. 18 (2009), no. 6, 1875-1916. · Zbl 1187.46056 |
| [21] | Hansen, F.: Selfadjoint means and operator monotone functions. Math. Ann. 256 (1981), no. 1, 29-35. · Zbl 0461.47009 |
| [22] | Hartz, M. and Lupini, M.: The classification problem for operator algebraic varieties and their multiplier algebras. Trans. Amer. Math. Soc. 370 (2018), no. 3, 2161-2180. · Zbl 1460.03019 |
| [23] | Hay, D. M., Helton, J. W., Lim, A. and McCullough, S.: Non-commutative partial matrix con-vexity. Indiana Univ. Math. J. 57 (2008), no. 6, 2815-2842. · Zbl 1171.15020 |
| [24] | Helton, J. W., Klep, I. and McCullough, S.: The matricial relaxation of a linear matrix inequal-ity. Math. Program. Ser. A 138 (2013), no. 1-2, 401-445. · Zbl 1272.15012 |
| [25] | Helton, J. W., Klep, I. and McCullough, S.: Free convex algebraic geometry. In Semidefinite optimization and convex algebraic geometry, 341-405. MOS-SIAM Ser. Optim. 13, SIAM, Philadelphia, PA, 2013. · Zbl 1360.14008 |
| [26] | Helton, J. W., Klep, I. and McCullough, S.: The tracial Hahn-Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 6, 1845-1897. · Zbl 1457.14123 |
| [27] | Helton, J. W. and McCullough, S.: Convex noncommutative polynomials have degree two or less. SIAM J. Matrix Anal. Appl. 25 (2004), no. 4, 1124-1139. · Zbl 1102.47009 |
| [28] | Helton, J. W. and McCullough, S.: Every convex free basic semi-algebraic set has an LMI representation. Ann. of Math. (2) 176 (2012), no. 2, 979-1013. · Zbl 1260.14011 |
| [29] | Helton, J. W., McCullough, S. A. and Vinnikov, V.: Noncommutative convexity arises from linear matrix inequalities. J. Funct. Anal. 240 (2006), no. 1, 105-191. · Zbl 1135.47005 |
| [30] | Helton, J. W. and Merino, O.: Sufficient conditions for optimization of matrix functions. In Proceedings of the 37th IEEE Conference on decision and control, vol. 3, 3361-3365. IEEE Control Systems Society, 1998. |
| [31] | Jorswieck, E. and Boche, H.: Majorization and matrix-monotone functions in wireless com-munications. Foundations and Trends ® in Communications and Information Theory 3.6, Now Foundations and Trends, 2007. · Zbl 1124.94009 |
| [32] | Jury, M., Klep, I., Mancuso, M. E., McCullough, S. and Pascoe, J. E.: Noncommutative partial convexity via -convexity. J. Geom. Anal. 31 (2021), no. 3, 3137-3160. · Zbl 1475.46070 |
| [33] | Kaliuzhnyi-Verbovetskyi, D. S. and Vinnikov, V.: Singularities of rational functions and min-imal factorizations: the noncommutative and the commutative setting. Linear Algebra Appl. 430 (2009), no. 4, 869-889. · Zbl 1217.47032 |
| [34] | Kaliuzhnyi-Verbovetskyi, D. S. and Vinnikov, V.: Foundations of free noncommutative func-tion theory. Mathematical Surveys and Monographs 199, American Mathematical Society, Providence, RI, 2014. · Zbl 1312.46003 |
| [35] | Kanev, S., Scherer, C., Verhaegen, M. and De Schutter, B.: Robust output-feedback controller design via local BMI optimization. Automatica J. IFAC 40 (2004), no. 7, 1115-1127. · Zbl 1051.93042 |
| [36] | Klep, I. and Volčič, J.: Free loci of matrix pencils and domains of noncommutative rational functions. Comment. Math. Helv. 92 (2017), no. 1, 105-130. · Zbl 1365.15017 |
| [37] | Kraus, F.: Über konvexe Matrixfunktionen. Math. Z. 41 (1936), no. 1, 18-42. · JFM 62.1079.02 |
| [38] | Pascoe, J. E. and Tully-Doyle, R.: The royal road to automatic noncommutative real analyticity, monotonicity, and convexity. Preprint 2019, arXiv: 1907.05875. |
| [39] | Passer, B., Shalit, O. M. and Solel, B.: Minimal and maximal matrix convex sets. J. Funct. Anal. 274 (2018), no. 11, 3197-3253. · Zbl 1422.47021 |
| [40] | Passer, B. and Shalit, O. M.: Compressions of compact tuples. Linear Algebra Appl. 564 (2019), 264-283. · Zbl 07007601 |
| [41] | Paulsen, V.: Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2002. · Zbl 1029.47003 |
| [42] | Popescu, G.: Invariant subspaces and operator model theory on noncommutative varieties. Math. Ann. 372 (2018), no. 1-2, 611-650. · Zbl 1460.47005 |
| [43] | Salomon, G., Shalit, O. M. and Shamovich, E.: Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball. Trans. Amer. Math. Soc. 370 (2018), no. 12, 8639-8690. · Zbl 1454.46058 |
| [44] | Schützenberger, M. P.: On the definition of a family of automata. Information and Control 4 (1961), 245-270. · Zbl 0104.00702 |
| [45] | Skelton, R. E., Iwasaki, T. and Grigoriadis, K. M.: A unified algebraic approach to linear con-trol design. The Taylor & Francis Systems and Control Book Series, Taylor & Francis Group, London, 1998. |
| [46] | VanAntwerp, J. G. and Braatz, R. D.: A tutorial on linear and bilinear matrix inequalities. J. Process Control 10 (2000), no. 4, 363-385. |
| [47] | Volčič, J.: On domains of noncommutative rational functions. Linear Algebra Appl. 516 (2017), 69-81. · Zbl 1376.16022 |
| [48] | Volčič, J.: Matrix coefficient realization theory of noncommutative rational functions. J. Algebra 499 (2018), 397-437. · Zbl 1441.16030 |
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