Let $L$ and $R$ be $n \times n$ matrices. Consider the following minimization problem  \begin{equation} \mathbf{L} = \min_{L \geq \mathbf{0}}  \mu\|\mathbf{L}\|_* + \dfrac{1}{2\lambda}\|\mathbf{L-R}\|_F^2 \end{equation}     By $\mathbf{L}$ $\geq$ 0 , i mean that i wish the entries of $\mathbf{L}$ to be non negative. I know the solution can be obtained using the singular value thresholding of $\mathbf{R}$ when the non-negativity constraint on $\mathbf{L}$ is not present. But I can't figure out the change needed in order to satisfy the non-negativity constraint on $\mathbf{L}$. Will simply setting the negative entries of $\mathbf{L}$ to zero work? This link  is about $\mathbf{L}$ being positive definite; my question is about the entries of $\mathbf{L}$ being non-negative. Can someone please answer this?
Let $L$ and $R$ be $n \times n$ matrices. I am trying to solve the following minimization problem  $$ L^{k+1} := \arg \min_{L} \lambda \|L\|_{\ast} + \frac{1}{2\mu}\|L-R^{k}\|_{F}^{2} $$ $$ \mbox{s.t.}\ L \succeq 0 $$  where $\|\cdot\|_{\ast}$ and $\|\cdot\|_{F}$ denote the nuclear and Frobenius norms, respectively. Also, $k$ expresses the $k$th iteration when solving an optimization problem successively.  I don't know how to solve this problem. Would you tell me the way?
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I have been thinking about doing the number-theory research on my university (research university; huge mathematics department). I have been self-studying the NT by myself and also regularly attending the professional and graduate seminars on number theory but I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them about if I can do undergraduate research under them? If research is not possible (perhaps due to my lacking maturity), should I request of doing independent reading under them and later proceed with the research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask to them or do in my own?   As for my mathematical background, I have been taking Calculus II (computational) and discrete mathematics. I will be taking calculus III (vector calc.) on Summer, followed by Analysis I, Probability, Theoretical Linear Algebra on Fall 2015. 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Here is MWE:  % !TeX program = xelatex % !TeX encoding = utf-8 % !TeX spellcheck = ru_RU  \documentclass[12pt, a4paper]{article}  \usepackage{polyglossia} \setdefaultlanguage{russian} \setmainfont[Ligatures=TeX]{Times New Roman}  \usepackage[xindy={glsnumbers=false}, nonumberlist, nopostdot, nogroupskip]{glossaries} %\setglossarystyle{super} % uncomment for style "super" \makeglossaries  \begin{document}  \newacronym{bam}{ДАП}{Двунаправленная Ассоциативная Память} \newacronym{ai}{ИИ}{Искусственный Интеллект} \newacronym{am}{АП}{Ассоциативная Память} \newacronym{ann}{ИНС}{Искусственная Нейронная Сеть} \newacronym{an}{ИН}{Искусственный Нейрон} \newacronym{it}{ИТ}{Информационные Технологии} \glsaddall  \printglossaries  \end{document}   With the default style I have the following:  As you can see, the long names are not vertically aligned. 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