×

On the semidefinite representation of real functions applied to symmetric matrices. (English) Zbl 1282.90122

Summary: We present a new semidefinite representation for the trace of a real function \(f\) applied to symmetric matrices, when a semidefinite representation of the convex (or concave) function \(f\) is known. Our construction is intuitive, and yields a representation that is more compact than the previously known one. We also show with the help of matrix geometric means and a Riemannian metric over the set of positive definite matrices that for a rational exponent \(p\) in the interval \((0, 1]\), the matrix \(X\) raised to \(p\) is the largest element of a set represented by linear matrix inequalities. This result further generalizes to the case of the matrix \(A\sharp_p B\), which is the point of coordinate \(p\) on the geodesic from \(A\) to \(B\). We give numerical results for a problem inspired from the theory of experimental designs, which show that the new semidefinite programming formulation can yield an important speed-up factor.

MSC:

90C22 Semidefinite programming
62K05 Optimal statistical designs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alizadeh, F.; Goldfarb, D., Second-order cone programming, Math. Program., 95, 1, 3-51 (2003) · Zbl 1153.90522
[2] Andersen, E. D.; Jensen, B.; Jensen, J.; Sandvik, R.; Worsøe, U., Mosek version 6 (2009), Technical report, Technical Report TR-2009-3, MOSEK
[3] Ando, T., Concavity of certain maps on positive definite matrices and applications to hadamard products, Linear Algebra Appl., 26, 203-241 (1979) · Zbl 0495.15018
[4] Ando, T.; Li, C.; Mathias, R., Geometric means, Linear Algebra Appl., 385, 305-334 (2004) · Zbl 1063.47013
[5] Ben-Tal, A.; Nemirovski, A., Lectures On Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, vol. 2 (1987), Society for Industrial Mathematics
[6] Bhatia, R., Positive Definite Matrices (2008), Princeton University Press
[7] Bhatia, R.; Holbrook, J., Riemannian geometry and matrix geometric means, Linear Algebra Appl., 413, 2, 594-618 (2006) · Zbl 1088.15022
[8] Bini, D.; Iannazzo, B., Computing the Karcher mean of symmetric positive definite matrices, Linear Algebra Appl., 438, 4, 1700-1710 (2013) · Zbl 1268.15007
[9] Bini, D.; Meini, B.; Poloni, F., An effective matrix geometric mean satisfying the Ando-Li-Mathias properties, Math. Comp., 79, 269, 437-452 (2010) · Zbl 1194.65065
[10] Boyd, S.; Vandenberghe, L., Convex Optimization (2004), Cambridge University Press · Zbl 1058.90049
[11] Fedorov, V.; Wu, Y.; Zhang, R., Optimal dose-finding designs with correlated continuous and discrete responses, Stat. Med. (2012)
[12] Freund, R. M.; Ordóñez, F.; Toh, K., Behavioral measures and their correlation with ipm iteration counts on semi-definite programming problems, Math. Program., 109, 2-3, 445-475 (2007) · Zbl 1278.90447
[13] Laraki, R.; Lasserre, J. B., Computing uniform convex approximations for convex envelopes and convex hulls, J. Convex Anal., 15, 3, 635-654 (2008) · Zbl 1153.90011
[14] Löfberg, J., Yalmip: A toolbox for modeling and optimization in matlab, (Computer Aided Control Systems Design, 2004 IEEE International Symposium (2004), IEEE), 284-289
[15] Moakher, M.; Zéraï, M., The riemannian geometry of the space of positive-definite matrices and its application to the regularization of positive-definite matrix-valued data, J. Math. Imaging Vision, 40, 2, 171-187 (2011) · Zbl 1255.68195
[16] Papp, D., Optimal designs for rational function regression, J. Amer. Statist. Assoc., 107, 497, 400-411 (2012) · Zbl 1261.62072
[17] Pukelsheim, F., Optimal Design of Experiments (1993), Wiley · Zbl 0834.62068
[18] Sagnol, G., A class of semidefinite programs with a rank-one solution, Linear Algebra Appl., 435, 6, 1446-1463 (2011) · Zbl 1220.90084
[19] Sagnol, G., Computing optimal designs of multiresponse experiments reduces to second-order cone programming, J. Statist. Plann. Inference, 141, 5, 1684-1708 (2011) · Zbl 1207.62156
[20] Sagnol, G., Picos, a python interface to conic optimization solvers (2012), ZIB, Technical report 12-48
[22] Sturm, J. F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11-12, 625-653 (1999) · Zbl 0973.90526
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.