In one and two dimensions, all roads lead to Rome, but not in three dimensions. Specifically, given a random walk (equally likely to move in any direction) on the integers in one or two dimensions, ...

Here is R1, as computed in MATLAB: 1.0e+07 * -7.382605957465515 -9.599867106092937 -2.830412177259742 -0.000000000002830 -0.000000000002830 -1.230434326244253 -1.599977851015490 -0....

The whole point is that you do NOT include the constraint $U = uu^T$ . That constraint is non-convex. Instead you include the constraint that $Z_j$ is (positive) semi-definite. This constraint is ...

Because you say the losses are convex, I will presume that all $c_i \ge 0$, which means that max is used in a convex fashion. Given that, this problem can be formulated as a Linear Programming problem ...

Because the matrix KK is not PSD. It has minimum eigenvalue of -1. It is indefinite. Perhaps you are under the misapprehension that a symmetric matrix with all elements positive is PSD. As this ...

If A and B are known, then the right hand side is known, and must of necessity be symmetric positive semi-definite. Therefore, there is a solution C to the stated equation, and this is the (lower ...

You can try using SCS, either the direct or indirect solver. SCS uses first-order methods, and hence may be able to solve larger problems than second-order solvers such as SDPT3, SeDuMi, MOSEK, etc. ...

In SCP (a.k.a. SCA), at each outer iteration: Objective function is replaced by a convex approximation, not necessarily quadratic. Nonlinear inequality constraints are replaced by convex ...

This is easy to formulate in CVX, under MATLAB. A CVXPY solution, under Python, is similar. CVX code: cvx_begin sdp variable X(n,n) hermitian semidefinite minimize(norm_nuc(X-A)) X <= B cvx_end or ...

Look at section 5 of http://web.stanford.edu/group/SOL/papers/ggms74.pdf to see how to update QR decomposition when a row is added to the matrix you wish to factorize. You may find it not so easy to ...

If you want to maximize f(X) subject to specified constraints (if any), then minimize -f(X) subject to those same constraints. Your optimal objective function will be the negative of -f(X_optimal), ...

The Frobenius norm of a binary matrix is the square root of the number of non-zero elements. Let the point (0.,...0) be the origin, and let's say the vec'd binary matrix elements are the coordinates ...

Yes, it would be called the Simplex algorithm. An active set method for solving Quadratic Programming problems is often called a "Simplex algorithm" (which is as opposed to an Interior Point method).

There are many sparse matrices in Matrix Market A visual repository of test data for use in comparative studies of algorithms for numerical linear algebra, featuring nearly 500 sparse matrices ...

Lower level optimization problems being solved within a top or higher level algorithm are called subproblems. So the algorithm or routine to solve subproblems could be called a "subproblem solver". ...

In order for a QCQP to be convex, the quadratic terms needs be be convex. Linear terms are always convex. With regard to your specific problem, the objective function is convex because it is linear. ...

The constraint $\det(A) > 0$ is very unstable and absolutely horrible. Would you believe that on practical, non-contrived 6 by 6 symmetric matrices, changing one element of the matrix by 1e-15 can ...

You are looking for a Newton's method, as opposed to Quasi-Newton used by SNOPT. You can consider using KNITRO, which can handle any optimization problem type that SNOPT can, but has a greater number ...

@Federico Poloni 's fine answer states the impossibility of getting an exact yes/no answer using IEEE arithmetic. However, using interval arithmetic with outward rounding, it is possible to get a &...

The authors provide bounds on various things as an explicit function of the iterate $n$, for a generic, but finite, $n$. These bounds apply for finite $n$, not only in the limit as $n \rightarrow \...

First of all, move the objective to the constraints by using the epigraph formulation max t subject to log_det(A) $\ge$ t plus your other constraints. Section 6.2.3 of the Mosek Modeling ...

The error message is because you forgot the . before * before exp in psi2. Once you fix that, MATLAB will be none too happy about the singularity you get dividing by abs(r1-r2) in I and J. If (1./abs(...

FMINCON is not able to find a feasible point starting at what I think is the default value provided by YALMIP of all variables being zero vectors. Local solvers, such as FMINCON, may have trouble ...

The constraints are not convex. Consider the example below in which x1 and x2 are each vectors of 3 elements which satisfy the inequality in question, as shown. 0.5(x1 + x2) does not satisfy the ...

KNITRO has Python and MATLAB interfaces, among others. Think of it as an FMINCON replacement, but much better performing, and more expensive. https://www.artelys.com/en/optimization-tools/knitro#doc-...

Eigenvectors are only unique to within a scale factor (can be + or - scale factor). If $x$ satisfies $Ax=\lambda x$, and hence is an eigenvector of $A$ corresponding to eigenvalue $\lambda$, then any ...

This is just a standard convex Quadratic Programming (QP) problem, which quadprog, or any number of other QP solvers, can solve. There are a variety of algorithms for solving QPs, and I think you ...

If you are willing to minimize the 2-norm of the solution rather than minimizing the number of non-zeros, you can use the pseudo-inverse. To solve A*x = b in MATLAB, that would be pinv(A)*b. In your ...

Per http://www.nag.com/numeric/fl/manual/pdf/F08/f08msf.pdf , regarding the LAPACK routine in the NAG Library for complex singular values/singular vectors. (Not generalized, but I think the idea is ...

This may be more of a comment than an answer, but I can't comment. Yes on arbitrary precision. No on symbolic inverse, because at some point, you have to numerically evaluate it, and it is likely to ...