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SUMMARY:Matrix Inequalities with Matrix Unknowns - Professor Bill Helton (
 Mathematics Department\, UC San Diego)
DTSTART:20100416T130000Z
DTEND:20100416T140000Z
UID:TALK24218@talks.cam.ac.uk
CONTACT:Dr Ioannis Lestas
DESCRIPTION:Linear matrix inequalities LMIs are common in many areas: cont
 rol systems\, combinatorial optimization\, statistics\, etc. They often ha
 ve unknowns\n x= ( x_1\, ... \,x_n) with x_j scalars\, but in many problem
 s of control\, certainly the classical ones\, the unknowns enter naturally
  as matrices.\n\nThe talk treats several topics involving LMIs with matrix
  unknowns:\n\nA basic question in light of the fact that convexity\, a see
 mingly much weaker condition than being an LMI\, guarantees numerical succ
 ess is: \n_How much more restricted are LMIs than Convex MIs?_ \nIt turns 
 out that  scalar unknowns vs matrix unknowns makes a huge difference in th
 e answer.\n\nCan we transform a problem to being convex?\n\nLMI domination
 : L dominates  \\L means L(x) is positive definite implies \\L(x) is posit
 ive definite. Checking for domination numerically can be NP hard. However\
 , we relax the problem by insisting on domination whenever the unknowns x_
 j are matrices.\nThere is an elegant algebraic characterization of relaxed
  LMI domination and numerical solution is no longer NP hard. Roughly what 
 we observed is that this matrix relaxation corresponds exactly to a very n
 atural procedure in modern  in modern functional analysis.\n
LOCATION: Cambridge University Engineering Department\, LR6
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