Re: Optimization for a positive definite constraint
- From: spellucci@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (Peter Spellucci)
- Date: Thu, 16 Feb 2006 19:20:32 +0000 (UTC)
In article <dt2aoc$q5p$1@xxxxxxxxxxxx>,
Hiu Chung Law <antispam@xxxxxxxxxxxx> writes:
Let X be a n by n matrix.
The goal is to minimize a real-value function f(X)
subject to the constraint that X is symmetric, and X is positive definite.
(The function f is not defined for X that is not positive definite).
I am now treating this as an optimization problem with n^2 variables,
and the gradient and the Hessian can be computed.
The symmetric constraint can be handled easily because it is linear.
However, the constraint that X should be positive-definite is more difficult
to handle. Right now, I simply force my line-search procedure to reject
any X that is not positive definite. However, this does not seem to be
robust. I have also re-parameterized X by X = F * F', but the Hessian
with respect to F is much more nasty than the Hessian with respect to X.
I thought something related to how to handle bound constraint may help, but
I cannot find anything similar. Also, this is quite different from
semi-definite programming....
Do you have any advice on how to tackle this type of problem?
Thank you.
if f(X) is not sum_{i=1to n} x(i)*A(i) with given symmetric positive
semidefinite matrices A(i) (semidefinite progamming)
but rather a "general" smooth function
then the approach you already tried
X=F*F' F lower triangular with positive diagonal
seems to me the best one, because, if for
example you go the way of parametrizing by the eigenvalues,
X=U*diag(lambda(i))*U' , lambda(i) >=0
you will also have the U as unknowns and the nasty orthogonality constraints
U'*U = I . Although there are special tricks to deal with orthogonality
constraints (paper by alan edelman and coworkers) this is even more nasty.
cutting the step is a faulty approach.
can you tell more about your f?
hth
peter
.
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