I need to handle such a problem. Consider the following equation:
P-1(A+E)P=B
with A and B given and E optional.
How to find an optimal E to minimize both norm(E) and cond(P)? Or at least minimize norm(E)?
Or can you indicate any reference book on matrix optimization where I might find an answer by myself?
You can look into Optimal Designed Experiments. They deal with issues similar to yours. Since what you are asking is such a pain in the rear, statisticians use a randomly generated matrix, and augment it as needed, then they generate another matrix and so on..
Dear Prof. Ekstrom,
Thank you very much for your kind help!
Are you sure that my problem is really so difficult that there exist no analytical algorithms at all? If so, then your approach should be the best.
Can you help me to confirm the book you mentioned? Is it this:
http://www.amazon.com/Optimal-Design-Experiments-Study-Approach/dp/0470744618
The methodology I'm referring to was developed about 140-150 years ago. What happened was the algorithms they developed were so difficult that no one really used them until recently.
If I want to create a D-Optimal design, I need to generate a matrix X and manipulate this matrix until you "maximize" the determinate of X'X. IF you look at how some of the different software programs perform this, there is a bit of disagreement. One company forces you to create a much larger design and chooses the optimal design from those design points. Another restricts the optimal values and uses those restricted design points to choose from. A third allows any value out to 3-4 decimal places, and chooses from those. What happens is that I can find 3 different "optimal" designs, using the same criteria. If I rerun the same optimal criteria on the same software, I will get slightly different optimal designs.
Brad Jones book is a good one for understanding the theory and application of Optimal Designs. The computer algorithms his software uses is proprietary. So, there will be some basic ideas on how to create these designs. They do not go into specifics.
I wish I could recommend a textbook that would answer your question completely. However, I haven't seen/used one. I would hope that a book from SIAM on applied matrix methods would be of help. If your matrix is small enough, implementing something by hand would be possible. But, for a larger matrix, you'll spend hours working at it.
One of the things that has always bothered me about some of the algorithms as taught in pure math classes is how profs make it look look so "simple". Then you spend hours doing it by hand and second doing it approximately with a computer.
Genetic algorithm maybe? It would consume time but eventually, put you in the right place...
as yair tamar said by defining a good fitness function which fits best with your problem you can find the answer.
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