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Convex analysis on the Hermitian matrices
 SIAM Journal on Optimization
, 1996
"... There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions ..."
Abstract

Cited by 42 (17 self)
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There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues. A new approach to this characterization is given, via a simple Fenchel conjugacy formula. We then apply this formula to derive expressions for subdifferentials, and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables. Analogous results hold for Hermitian matrices. Key Words: convexity, matrix function, Schur convexity, Fenchel duality, subdifferential, unitarily invariant, spectral function, positive semidefinite programming, quasiNewton update. AMS 1991 Subject Classification: Primary 15A45 49N15 Secondary 90C25 65K10 1 Introduction A matrix norm on the n \Theta n complex matrices is called unitarily inv...
The Convex Analysis of Unitarily Invariant Matrix Functions
, 1995
"... this paper is to give a simple, selfcontained approach to this problem, giving back the subdifferential formula for (1.2) in [13] for example. Our idea will be to generalize von Neumann's result somewhat by asking which convex functions (rather than simply norms) are unitarily invariant: appropriat ..."
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Cited by 28 (2 self)
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this paper is to give a simple, selfcontained approach to this problem, giving back the subdifferential formula for (1.2) in [13] for example. Our idea will be to generalize von Neumann's result somewhat by asking which convex functions (rather than simply norms) are unitarily invariant: appropriately, the key idea will be a Fenchel conjugacy formula analogous to von Neumann's polarity formula (1.1): (f ffi oe)
Group Invariance and Convex Matrix Analysis
 SIAM J. Matrix Anal. Appl
, 1995
"... Certain interesting classes of functions on a real inner product space are invariant under an associated group of orthogonal linear transformations. This invariance can be made explicit via a simple decomposition. For example, rotationally invariant functions on R 2 are just even functions of t ..."
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Cited by 18 (7 self)
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Certain interesting classes of functions on a real inner product space are invariant under an associated group of orthogonal linear transformations. This invariance can be made explicit via a simple decomposition. For example, rotationally invariant functions on R 2 are just even functions of the Euclidean norm, and functions on the Hermitian matrices (with trace inner product) which are invariant under unitary similarity transformations are just symmetric functions of the eigenvalues. We develop a framework for answering geometric and analytic (both classical and nonsmooth) questions about such a function by answering the corresponding question for the (much simpler) function appearing in the decomposition. The aim is to understand and extend the foundations of eigenvalue optimization, matrix approximation, and semidefinite programming. 2 1 Introduction Why is there such a strong parallel between, on the one hand, semidefinite programming and other eigenvalue optimizat...
On Approximation Problems With ZeroTrace Matrices
, 1994
"... In this paper we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms. We deal with special cases of approximation of a matrix by zerotrace matrices. Moreover, some characterizations of zerotrace matrices are given by means of matri ..."
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Cited by 2 (0 self)
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In this paper we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms. We deal with special cases of approximation of a matrix by zerotrace matrices. Moreover, some characterizations of zerotrace matrices are given by means of matrix approximation problems.  1. INTRODUCTION Let A = [a ij ] 2 C n\Thetan be a complex matrix. The trace of A is equal to tr(A) = X j a jj : It is wellknown that tr(A) = 0 if and only if A is a commutator, that is, A = XY \Gamma Y X for some matrices X and Y . In this paper we consider some approximation problems, involving zerotrace matrices, with respect to an arbitrary unitarily invariant norm jj \Delta jj. A norm jj \Delta jj is unitarily invariant if jjUAjj = jjAV jj = jjAjj for all unitary matrices U and V . The most popular unitarily invariant norms are the c p ...
Design of a Class of Multirate Systems Using a Maximum Relative l²Error Criterion
, 1996
"... . A criterion for designing the class of multirate systems for rate#changing is presented. This criterion arises from a model#matching perspective with maximum relative # ..."
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Cited by 1 (0 self)
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. A criterion for designing the class of multirate systems for rate#changing is presented. This criterion arises from a model#matching perspective with maximum relative #