User todpole3 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:44:36Z http://mathoverflow.net/feeds/user/27302 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109844/how-to-find-a-matrix-closest-to-a-given-one-under-certain-constraints How to Find a Matrix Closest to a Given One Under Certain Constraints todpole3 2012-10-16T19:26:37Z 2013-01-25T06:22:00Z <p>I was reading a paper about BFGS and met the following problem:</p> <p>$\min_B \|B-B_k\|$, s.t. $B=B^{\top}, Bs_k=y_k, s_k^{\top}y_k>0$ and $B$ is positive definite. Here $B_k$ is a symmetric positive definite $n\times n$ matrix, $y_k,s_k$ are all $n\times 1$ vectors. $B_k, y_k,s_k$ are all known.</p> <p>The norm used to measure the closeness between matrices is defined to be the weighted Frobenius norm, $\|A\|=\|W^{\frac{1}{2}}AW^{\frac{1}{2}}\|_F$.</p> <p>The paper then gives the solution of this minimization problem as $B^* = (I-\rho_ky_ks_k^{\top})B_k(I-\rho_ks_ky_k^{\top})+\rho_ky_ky_k^{\top}$, where $\rho_k=\frac{1}{y_k^{\top}s_k}$, and calls it the Davidon-Fletcher-Powell formula.</p> <p>I didn't have any clues about how to solve this optimization in closed form. Also, materials explaining the DFP formula seems to be ridiculously difficult to find. Could any one help?</p> <p>Even any hints on solving matrix norm optimization problem would help, thanks!</p> http://mathoverflow.net/questions/109845/what-dose-rank-two-modification-w-r-t-matrix-mean What dose "Rank-two Modification w.r.t Matrix" Mean? todpole3 2012-10-16T19:36:13Z 2012-10-16T22:26:42Z <p>I came over "rank-one modification of $A$", "rank-two modification of $B$" in my readings and want to know what does "rank-N modification" w,r,t a matrix mean?</p> http://mathoverflow.net/questions/109844/how-to-find-a-matrix-closest-to-a-given-one-under-certain-constraints/109863#109863 Comment by todpole3 todpole3 2012-10-17T02:15:24Z 2012-10-17T02:15:24Z seems not. Wikipedia directly gives the solution and didn't derive it. Now I can somehow see this could follow from solving a Lagrange multiplier problem. But the form of the equation is crazy...