Matrix Functions and Weighted Centers for Semidefinite Programming

In this paper, we develop various differentiation rules for general smooth matrix-valued functions, and for the class of matrix convex (or concave) functions first introduced by Löwner and Kraus in the 1930s. For a matrix monotone function, we present formulas for its derivatives of any order in an integral form. Moreover, for a general smooth primary matrix function, we derive a formula for all of its derivatives by means of the divided differences of the original function. As applications, we use these differentiation rules and the matrix concave function log X to study a new notion of weighted centers for Semidefinite Programming (SDP). We show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the derivative formulas can be used in the implementation of barrier methods for optimization problems involving nonlinear but convex matrix functions.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[3]  Jein-Shan Chen.  The convex and monotone functions associated with second-order cone , 2006 .

[4]  Chek Beng Chua.  A New Notion of Weighted Centers for Semidefinite Programming , 2006, SIAM J. Optim..

[5]  T. Georgiou.  Relative Entropy and Moment Problems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[6]  Paul Tseng,et al.  Analysis of nonsmooth vector-valued functions associated with second-order cones , 2004, Math. Program..

[7]  Zhi-Quan Luo,et al.  Transceiver optimization for block-based multiple access through ISI channels , 2004, IEEE Transactions on Signal Processing.

[8]  Shuzhong Zhang.  On Conically Ordered Convex Programs , 2004 .

[9]  Paul Tseng,et al.  Non-Interior continuation methods for solving semidefinite complementarity problems , 2003, Math. Program..

[10]  Paul Tseng,et al.  Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems , 2003, SIAM J. Optim..

[11]  Defeng Sun,et al.  Semismooth Matrix-Valued Functions , 2002, Math. Oper. Res..

[12]  Adrian S. Lewis,et al.  Twice Differentiable Spectral Functions , 2001, SIAM J. Matrix Anal. Appl..

[13]  Shuzhong Zhang,et al.  On weighted centers for semidefinite programming , 2000, Eur. J. Oper. Res..

[14]  C. Roos,et al.  Theory and algorithms for semidefinite programming , 2000 .

[15]  R. Saigal,et al.  Handbook of semidefinite programming : theory, algorithms, and applications , 2000 .

[16]  Timothy F. Havel,et al.  Derivatives of the Matrix Exponential and Their Computation , 1995 .

[17]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[18]  E. T. Jaynes,et al.  Theory of Gyromagnetic Effects and Some Related Magnetic Phenomena , 1962 .

[19]  R. Feynman.  An Operator calculus having applications in quantum electrodynamics , 1951 .

[20]  F. Kraus.  Über konvexe Matrixfunktionen , 1936 .

[21]  Karl Löwner.  Über monotone Matrixfunktionen , 1934 .