# Tagged Questions

Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

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### References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
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### Why this problem is unbounded, even though it's dual is feasible?

In this paper, An Efficient Inexact ABCD Method for Least Squares Semidefinite Programming,page 2, there is a problem called, (P): \begin{align} P: &\min_{X,s} && ‎ \frac{1}{2} ‎\Vert X-G\...
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### On Boyd et al.'s convergence analysis of ADMM: Why do we need the convexity assumption?

Please refer to Boyd et al.'s convergence analysis of ADMM (Chapter 3 and Appendix A). My question is: Why do we need $f$ and $g$ to be convex? I don't see the need of this assumption. If the ...
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### Converting generic linear problems into their dual

I'm revising how to do dual problems in linear algebra. I'm very weak in Linear programing but I struggle to cope with the topic during lectures and assignements. I have to convert the following ...
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### How to deal with an $xy\le 1$ constraint?

I have to solve the following optimization problem: \begin{align*} \min_{x,y} &\{-x-y\} \\ \text{such that} \\ y &\ge 3 \\ y &\le 30 \\ x &\ge 0 \\ xy &\le 1 \\ \end{align*} ...
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### Convexify $x\le a+by^2$

I have the following non-convex constraint: $$x\le a+by^2\quad\text{where}\quad a,b>0,\,y\in[0,y_{max}]\text{ and }a\approx by_{max}^2$$ On a drawing, it looks something like this: The above ...
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### Coordinate descent with equality and inequality constraints

I have an intuitive understanding of why the simple method of coordinate descent does not work with linearly coupled constraints such as; $$\min_x\sum_if_i(x_i)$$ $$s.t.$$ $$Ax=b$$ If we try to ...
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### How to compute norm bound error in robust approximation

I am reading convex optimization, and I am little confused about the following two prolems in norm bound error of robust approximation. How to compute $\{\|\bar{A}X-b+Ux\| | \|U\|\le a\}$ ? For the ...
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### Can the constrained optimization problem (1) be transformed into the unconstrained form (2)

(1) \label{constrained} \begin{array}{cl} \arg \min \limits_{\mathcal{C}_k} & \text{rank}(\mathcal{C}_k)\\ \mathrm{s.t.} & \mathcal{E}(\phi_{j}^{k})\le \epsilon \end{...
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### general framework for proof of convexity in least square problem

Let $f(x_i,\theta) : X \times \Theta \rightarrow Z$ be a parametric function with parameter $\theta$ that we wish to fit to set of samples $S =\{(x_i,z_i)\}_{i=1}^N$, where $(x_i, z_i) \in X \times Z$....
Compute the minimal value of $$x_1 + 2x_2 + 3x_3$$ when $x_1$, $x_2$, $x_3$ satisfy $$x_1 − 2x_2 + x_3 = 4$$ $$−x_1 + 3x_2 = 5$$ and $$x_1 \ge 0, \qquad x_2 \ge 0, \qquad ... 1answer 50 views ### Convex set: extreme points and distance to the origin I'm fairly sure the following is true, although I wouldn't mind being proven wrong. If true, I would like to see an elegant proof, as my attempts are kind of messy. Let K\subset\mathbb R^2 be a ... 0answers 14 views ### Equality constraints into inequalities constraints through elimination I read here in Section 10.1.2 of this text that a way to eliminate linear equality constraints of the type$$Ax = b in convex optimization problems is to parameterize the related affine space as a ...
(1) $$\label{constrained} \begin{array}{cl} \arg \min \limits_{x} & \|Ax-b\|_2\\ \mathrm{s.t.} & \|x\|_1\le \epsilon \end{array}$$ (2) \label{...