User i j - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:32:39Z http://mathoverflow.net/feeds/user/11773 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75730/finding-smallest-ellipsoid-that-circumscribes-over-intersection-of-two-ellipsoids Finding smallest ellipsoid that circumscribes over intersection of two ellipsoids that do not have common center I J 2011-09-18T05:46:58Z 2012-11-11T10:32:25Z <p>Does this already exist in literature? The closest Ive been able to find is circumscribe intersection of two ellipsoids with a common center by W. Kahan (http://www.cs.berkeley.edu/~wkahan/Ellipint.pdf).</p> <p>I am looking for a method to circumscribe an ellipsoid over the intersection of two ellipsoids. The ellipsoid do not have a common center.<br> PS: We can assume that the ellipsoids always intersect and they are full dimensional ellipsoids (not enclosed in a subspace). However, the ellipsoids can be infinite cylinders (if the matrix W for (x-c)^TW(x-c) is not invertible).</p> http://mathoverflow.net/questions/79301/java-library-for-sdp Java library for SDP I J 2011-10-27T19:32:09Z 2011-10-28T08:04:33Z <p>People who frequently code semi definite programs, is there any java library for solving sdps? I have tried my luck but all I can find is C/C++ libraries or matlab toolboxes. I can write wrappers to call C from Java but will be much useful if there is indeed a java library.</p> http://mathoverflow.net/questions/66220/cayley-transform-for-non-square-matrices cayley transform for non-square matrices I J 2011-05-27T20:27:05Z 2011-08-17T17:56:24Z <p>Hi,<br> I am optimizing a function over a matrix $U$, where $U \in \mathbb{R}^{m \times n}$ and $U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint $U^TU = 1$ with lagrange multipliers would lead to higher time for convergence. I instead want to run simple gradient based optimization. So, I wanted to encode the orthogonality constraint into the input variable itself. What I mean by that is, for example, if $U$ was a square matrix then by Cayley transform for a skew-symmetric matrix $A$ we have: $U = (I-A)(I+A)^{-1}$ -- and then I can use unconstrained optimization with a lower dimensionality of input variables.</p> <p>So my question is whether there is an extension of the Cayley transform to non-square matrices of type $U^TU = I$ where $U \in \mathbb{R}^{m \times n}$.</p> <p>Thanks<br> I.J</p> http://mathoverflow.net/questions/79301/java-library-for-sdp/79363#79363 Comment by I J I J 2011-10-28T16:08:15Z 2011-10-28T16:08:15Z Unfortunately Mosek does not solve semi definite programming problems (<a href="http://docs.mosek.com/kb/Can_MOSEK_solve_semi-definite_optimizati.html" rel="nofollow">docs.mosek.com/kb/&hellip;</a>). I've decided to write a java interface to CSDP since it looks certain that there are no existing java interfaces. http://mathoverflow.net/questions/79301/java-library-for-sdp Comment by I J I J 2011-10-27T20:59:34Z 2011-10-27T20:59:34Z @Igor in reply to (c): Harsh words! Well it depends where you are coming from. Sometimes you have to use someone's code that is only in java and sometimes you are willing to take a hit in perf for design reasons. Lets not get into that. About the java interface with C: I guess you are talking about JNI, I am aware of that. http://mathoverflow.net/questions/79301/java-library-for-sdp Comment by I J I J 2011-10-27T20:19:14Z 2011-10-27T20:19:14Z @Igor: Your angry birds example does not have any intersection with MO. My post is not a math research question but chances that someone in MO community might have encountered this are higher than in SO. Anyways, feel feel to vote it down - let the votes decide. http://mathoverflow.net/questions/79301/java-library-for-sdp Comment by I J I J 2011-10-27T19:43:40Z 2011-10-27T19:43:40Z I meant to say &quot;small number of ppl at stackoverflow&quot;. http://mathoverflow.net/questions/79301/java-library-for-sdp Comment by I J I J 2011-10-27T19:42:35Z 2011-10-27T19:42:35Z I tried stackoverflow prior to this. There is a very small number of people who use things other than cplex, so it wasn't fruitful. http://mathoverflow.net/questions/75730/finding-smallest-ellipsoid-that-circumscribes-over-intersection-of-two-ellipsoids/75772#75772 Comment by I J I J 2011-09-18T17:46:47Z 2011-09-18T17:46:47Z Ah, that's a good point. I'll change tight to mean the smallest http://mathoverflow.net/questions/75730/finding-smallest-ellipsoid-that-circumscribes-over-intersection-of-two-ellipsoids Comment by I J I J 2011-09-18T17:11:50Z 2011-09-18T17:11:50Z @Suvrit: Ah, we can assume that the ellipsoids intersect and they are &quot;full&quot; ellipsoids (not embedded in a subspace). @Joseph: But I do not want to compute the intersection itself which would be quite expensive. Also, this is the link to the W. Kahan paper I talked about in my question: <a href="http://www.cs.berkeley.edu/~wkahan/Ellipint.pdf" rel="nofollow">cs.berkeley.edu/~wkahan/Ellipint.pdf</a> http://mathoverflow.net/questions/74016/minkowski-sum-of-a-vector-subspace-and-a-geometric-object-like-an-sphere-ellipse Comment by I J I J 2011-09-02T00:34:32Z 2011-09-02T00:34:32Z I realize that my question is more computational than pure math.I need 'efficient' intersections of unions of such objects. If $A_1$, $A_2$, etc. are the &quot;objects&quot; (Minkowski sums) then I want to compute an efficient intersections of their unions. That is I want to compute $(A_1 \cup A_2) \cap (B_1 \cup B_2)$. I don't want to implement this intersection as $(A_1 \cap B_1) \cup (A_1 \cap B_2) \cup (A_2 \cap B_1) \cup (A_2 \cap B_2)$ since with increased size of the unions, this would blow up exponentially. I can see that there is a problem here -basically the union can be a disjoint set. http://mathoverflow.net/questions/74016/minkowski-sum-of-a-vector-subspace-and-a-geometric-object-like-an-sphere-ellipse Comment by I J I J 2011-08-30T05:04:40Z 2011-08-30T05:04:40Z Agree with both of you. My question was whether such an object is well defined algebraically, so that I can compute intersections of multiple such objects. I know that the Minkowski sum is an algebraic definition of the space in itself but was looking for something more concrete than that so that I can actually compute intersections easily. http://mathoverflow.net/questions/66220/cayley-transform-for-non-square-matrices/73083#73083 Comment by I J I J 2011-08-23T05:14:56Z 2011-08-23T05:14:56Z Hi Juliano, I used conjugate gradient on Stiefel manifolds. Your suggestion of writing any point on the Stiefel manifold as product of m x m unitary matrix with a fixed m x n Y_0 is very neat. cheers http://mathoverflow.net/questions/64648/preconditioning-regularizing-ill-conditioned-overdetermined-system Comment by I J I J 2011-05-11T19:12:28Z 2011-05-11T19:12:28Z $A$ is a dense matrix. Thanks http://mathoverflow.net/questions/64648/preconditioning-regularizing-ill-conditioned-overdetermined-system Comment by I J I J 2011-05-11T17:45:17Z 2011-05-11T17:45:17Z I use the svd instead of taking the inverse of $A^TA$. Sure, i will just post this in math.stackexchange.com. Thanks http://mathoverflow.net/questions/50180/monte-carlo-integration-error-bounds/50212#50212 Comment by I J I J 2010-12-23T22:31:09Z 2010-12-23T22:31:09Z I did see the bounded variation formula in one variable. I too had my doubts whether the inequality is useful in higher dimensions. It seems to me that probably there are no useful bounds on the error in monte carlo sampling if you do not have something like a hypercube on which you are integrating the function. http://mathoverflow.net/questions/50180/monte-carlo-integration-error-bounds/50212#50212 Comment by I J I J 2010-12-23T04:03:56Z 2010-12-23T04:03:56Z Thanks. Can the variation of the function be computed over a convex set in general? http://mathoverflow.net/questions/50180/monte-carlo-integration-error-bounds/50191#50191 Comment by I J I J 2010-12-22T23:20:40Z 2010-12-22T23:20:40Z Gerry: Edited the question.