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                Department of Mathematical Sciences                       
                   The Johns Hopkins University                           
                                                                        
                              SEMINAR                                
                                                                        
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Jie Sun                                           March 8, 2001
Business School                                   304 Whitehead Hall
National University of Singapore                  NO PRESEMINAR
(visiting Massachusetts Institute of Technology)  Refreshments: 3:30 p.m.
                                                  Seminar: 4:00 p.m.

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                SEMISMOOTH MATRIX-VALUED FUNCTIONS

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                             ABSTRACT

Recent development in semidefinite optimization provides an interesting
source for the study of semismooth matrix-valued functions.  For example,
let  X  and  Y  be symmetric real matrices and let  [X - Y]_+  be the
projection of  X - Y  onto the cone of positive semidefinite matrices.  It
can be shown that the complementarity condition

X, Y \succeq 0,   X Y = 0

is equivalent to the matrix-valued equation

X - [X - Y]_+ = 0,

where  X, Y \succeq 0  means that both  X  and  Y  are positive
semidefinite.  Thus, if the function  [X - Y]_+  is semismooth (definition
will be given in the talk), then it is possible to design a Newton method
for the matrix complementarity problem.  The speaker will give a brief
introduction to the development of Newton's method for semismooth equations
and present some recent results on semismooth matrix-valued equations.

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