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I have got a quadratic form as following:

F(w) = g^T * w + (1/2)* w^T * B * w

Here T is the matrix transpose. Someone said the if p is a solution to this quadratic form then p is a solution to an equation of the form:

(B + aI) * p = -g

Here I is the identity matrix.

May I ask why please? Thanks a lot!

I am dealing with tomography inverse problem by using trust-region optimization now. I compared my result with other optimization algorithms found that trust-region results are much better. May I ask if the trust-region optimization subproblem contains a regularization already itself please? If it contains regularization itself and how does this related to Tikhonov regularization please?

By the way, I am using Matlab fminunc to implement the large scale trust-region Newton algorithm.

Thanks a lot and I am looking forward to your reply!

I met a problem about finding the optimization of some function. I used the Trust-Region Newton and Quasi-Newton methods for the problem; however, with different initial guesses I sometimes got the local minimums. May I ask how to get out the trap of the local minimums please?

I may try the Radom Walk method but it seems not be a good one. Any other ideas please reply! Thanks a lot!