On the semidefinite representation of real functions applied to symmetric matrices

Abstract 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 ♯ 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.

[1]  Erling D. Andersen,et al.  The homogeneous and self-dual model and algorithm for linear optimization. MOSEK Technical report: TR-1-2009. , 2009 .

[2]  Kim-Chuan Toh,et al.  Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems , 2007, Math. Program..

[3]  Guillaume Sagnol,et al.  Optimal monitoring in large networks by Successive c-optimal Designs , 2010, 2010 22nd International Teletraffic Congress (lTC 22).

[4]  Guillaume Sagnol,et al.  A class of semidefinite programs with rank-one solutions , 2009, 0909.5577.

[5]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[6]  D'avid Papp,et al.  Optimal Designs for Rational Function Regression , 2010, 1009.1444.

[7]  Rida Laraki,et al.  Computing uniform convex approximations for convex envelopes and convex hulls , 2008 .

[8]  Federico Poloni,et al.  An effective matrix geometric mean satisfying the Ando-Li-Mathias properties , 2010, Math. Comput..

[9]  R. Bhatia,et al.  Riemannian geometry and matrix geometric means , 2006 .

[10]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[11]  Valerii Fedorov,et al.  Optimal dose‐finding designs with correlated continuous and discrete responses , 2012, Statistics in medicine.

[12]  Dario Bini,et al.  Computing the Karcher mean of symmetric positive definite matrices , 2013 .

[13]  Guillaume Sagnol,et al.  Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming , 2009, 0912.5467.

[14]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[15]  Maher Moakher,et al.  The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data , 2011, Journal of Mathematical Imaging and Vision.

[16]  F. Pukelsheim.  Optimal Design of Experiments , 1993 .

[17]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[18]  Chi-Kwong Li.  Geometric Means , 2003 .

[19]  T. Andô.  Concavity of certain maps on positive definite matrices and applications to Hadamard products , 1979 .

[20]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[21]  R. Bhatia.  Positive Definite Matrices , 2007 .