# Tagged Questions

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

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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 $(x_i, z_i) \in X \times Z$....
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### Subdifferential optimality conditions

I need help with subdifferential optimality. Let $f(x_1, x_2)=x_1^2 + x_2^2 + |x_1 -x_2 - y|$. Find: \begin{align} \min_{x_1, x_2} f(x_1, x_2) \end{align} This is convex, so must have unique ...
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### Be C a matrix n x n positive semi definite. proof x'Cx is convex and sqrtroot(x'Cx) is convex.

Hi I have a homework from optimization and I want to know how to do the following exercise: Be C a matrix n x n positive semi definite. proof that: (1)$x^tCx$ is convex. (2)$\sqrt{x^tCx}$ is convex. ...
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### Convert Quadratically constrained basis pursuit to LASSO

The Quadratically constrained basis pursuit is to solve \begin{align} \hat{\boldsymbol{x}} &= \arg\min \|\boldsymbol{x} \|_1 \\ s.t. & \| \boldsymbol{Ax} - \boldsymbol{y} \|_2^2 < \eta \...
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### Existence of unique maximizer in R^n

This sounds like a very basic question, but I have a hard time pinpointing the necessary and sufficient conditions... Let $f : \mathbf{R}^n \to \mathbf{R}$ be a function. I want to prove that there ...
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### Properties of convex function with Lipshitz continuous gradient (Prof. Nesterov's textbook)

I am reading the Prof. Nesterov's textbook: Introductory lectures on convex optimization - a basic course p.57 I have problem in the following: My ...
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### Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems I am confused about the green part of the following: $\\$ ...
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### How to prove this function is quasi-concave? [closed]

Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?
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### Convex hull of halfspace and point is not a polyhedron

Let $S=conv(H \cup\{x\} )$ denote the convex hull of $H \cup\{x\}$ where $H \subset \Bbb{R}^n$ is a halfspace and $x\in\Bbb{R}^n, x\notin H$. I need to prove that $S$ is not a polyhedron and my ...
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### Can we solve minimax in this way?

I am working to use proximal operators for solving a minimax optimization problem. It is known that if you use alternative optimization, the algorithm cycles, see an answer to this question convex-...
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### How can I solve this as an optimization problem?

I would like to find x such that (Ax).^2 + (Bx).^2 == I (using Matlab syntax). A, B are matrices and I is a vector, all with real values. The number of equations is less than the number of variables, ...
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### Transform a nonconvex problem into a convex problem using perspective function

Suppose I have the problem $$\text{minimize } f_0(x)\\ \text{subject to } tf_1(x) \leq r$$ with variables $t,x \in \mathbb{R}$ and $f_0, f_1$ are convex. The constraint is not convex, so I was ...
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### Can we say anything about the minimum of a perspective function compared to that of the original function?

Given convex function $f(x)$, its perspective function is $g(x,t) = tf(x/t), t>0$ is also convex. Is the minimum of $g$ over $(x,t)$ always less than (or larger than) the minimum of $f(x)$? Note ...
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### Convex function and convex optimization

I would like to ask something about convex function and convex model. For example, the function $f(x,y)=\frac{x^2}{y}$ is convex when $x\geqslant0$ and $y>0$. For a convex model (minimization), ...
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### Least squares with multiple linear constraints

The method of direct elimination can be used to solve the constrained least squares problem $$\min_{\mathbf{x}}\left\Vert \mathbf{Ax}-\mathbf{b}\right\Vert _{2}$$ \begin{...
Given a three-order tensor $\mathcal{Y}$, our aim is to find a tensor $\mathcal{X}$ to approximate it and $\mathcal{X}$ should satisfy the following property: $\mathcal{X}$ can be well approximated ...
It is seen in Boyd's book on Convex Optimization book that to show a function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ is quasiconvex, it is enough to show that $f$'s restriction to a line is ...