Covering Number of a Positive Semidefinite Cone (Approximate the Objective of a SDP)

I was wondering what the covering number of a positive semidefinite cone is. Consider the semidefinite optimization program \begin{align} \max\langle \mathbf{C}, \mathbf{X} \rangle~~\text{subject to}~~ \langle \mathbf{A}_i, \mathbf{X} \rangle \leq b_i, i= 1,\ldots, m, \mathbf{X} \succeq \mathbf{0}. \end{align} Is it possible to find finite number of elements $\{ \mathbf{V}_k \}_{k=1}^N$ in the SDP cone \begin{align} \left\{ … Read more

Stochastic subgradient descent almost sure convergence

I was reading up on stochastic subgradient descent, and most sources i could find via google search give quick proofs on convergence in expectation and probability, and say that proofs of almost sure convergence exist. Can anyone point me to a proof of this claim? I would be especially interested in proving the following claim … Read more

Using Linear Programming as an iterative procedure

Suppose, we have a linear program and an optimal solution to it. Suppose now, we get a new constraint. We want to obtain an optimal solution to the given linear program extended by that new constraint. Is it possible to use the calculated optimal solution to do it in some clever way? Obviously, if the … Read more

Non-singularity of a series of matrices

Let A1, A2 be n×n real matrices. Suppose that A1 and A2 are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let B1, B2 be two n×m real matrices of full column rank (i.e. rankB1=rankB2=m) and define R1:=[B1|A1B1|A21B1|⋯|An−11B1] R2:=[B2|A2B2|A22B2|⋯|An−12B2] (In control theory the above-defined matrices are called reachability … Read more

Optimal control of SDEs

I’ve set up a system of stochastic differential equations that I’d like to control. I’m new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I’m not certain if I’ve set this up correctly. If someone could let me know if I’ve not completely bungled it and how to set it up … Read more

How to promote a blog?

Math behind might be interesting. Quite recent bloggingg activity might have interesting math model. The point is that bloggers compete for subscribers and at the same time cooperate gaining subscribers from partner’s blogs. Question is about existence of something like J.Nash’s equilibrium strategy which should balance competition/cooperation. Setup: There are inet blogs, each blog has … Read more

Can a nonlinear dynamical system be rewritten in terms of constraints?

My question is based on thoughts after reading to a specific section in the paper “On Contraction Analysis for Nonlinear Systems” by W. Lohmiller and JJ. Slotine, Section 4.2 Constrained Systems. Those interested in answering my question should probably read this section first. Question Suppose I have an n dimensional nonlinear dynamical system x′(t)=f(x)x∈Rn with … Read more

Least squares with matrix product constraints

I am encountering a constrained LS problem with the following structure: minQ M∑i=1||QiXi−Yi||2F s.t.  QMQM−1⋯Q1=I, where Qi,Xi,Yi∈Rn×n are full rank. I understand this can be a very hard problem. I just want to know if there is a numerically converging algorithm for this type of problem by any chance. Answer AttributionSource : Link , Question Author : … Read more

Looking for an electronic copy of Lebeau’s paper

I would like to know if anyone has an electronic copy of the paper “Gilles Lebeau – Contrôle De L’Équation De Schrödinger”? This article appeared in Journal de Mathématiques Pures et Appliquées, Vol. 71, (1992), no. 3, 267–291. The website of the Bibliothèque nationale de France – Gallica has all volumes all volumes published by … Read more

Minimizing the largest eigenvalue of matrix product

Let A\in\mathbb{C}^{m\times n}, B\in\mathbb{C}^{n\times k}, C\in\mathbb{C}^{k\times m} be given complex matrices. The objective of the optimization problem is \begin{equation} \mathop {\arg \min }\limits_X \lambda_{\max} \left( (A + BXC)(A + BXC)^H \right), \end{equation} where X\in\mathbb{C}^{k\times k} is a matrix with |x(i,j)|<1 for all i,j \in 1, 2,\dots,k? Answer AttributionSource : Link , Question Author : hichem … Read more