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

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Is this function convex or non-convex? How do you decide?

The problem is: find $$\min⁡ \mathrm{P}\left[{\log(1+p||H^H \mathbf{w}||^2)\over 1+p||G^H \mathbf{w}||^2}<R\right]$$ constraint to: $||\mathbf{w}||^2=1$ where $H$ and $G$ are matrices of ...
2
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1answer
128 views

Regarding Nesterov's smooth minimization

I am currently studying this Nesterov's paper for project purposes, and I am trying to figure out how the smoothing and the minimization algorithm works I have tried looking at the example ...
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21 views

What modification is this of the notion of Renyi divergence?

Given two probability distributions $P$ and $Q$ over the same outcome and event space (assume finite if needed) one defines their Renyi divergence as $D_\alpha (P \vert \vert Q) = \frac{1}{\alpha -1} ...
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1answer
636 views

convex hull function in matlab

Is there any way to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
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34 views

Why is the affine hull of the unit circle $\mathbb{R}^2$?

My question is addressed in Why is the affine hull of the unit circle $\mathbb R^2$? However, I am still confused. I thought that the affine of C in this case would be the interior of the circle. I ...
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27 views

Dot product - geometrical interpretation in convex analysis

I am studying a theorem on the characterization of solutions in nondifferentiable convex problems. Say that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and $f: \mathbb{R}^n \to ...
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1answer
25 views

Image restoration in matlab via PDE toolbox

I want to remove a noise for an image using matlab, when the observed image is $$f=u+v$$ where $u$ is the restored image (is the image i want recovered) and $v$ is the gaussian noise. To restore $u$, ...
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42 views

Concave optimization and corner solution

I have a optimization problem as follows: Assumptions: $f$ is an increasing and convex function on $R^+$ such that: $f(x): R^+\rightarrow R^+, \quad f(0)=0, \quad f'(x)\ge1,\quad f''(x)\ge 0 ...
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1answer
34 views

How to pivot to an adjacent vertex in simplex method

In the simplex method, we need to move from one vertex of the polyhedron to an adjacent one. Suppose the polyhedron is $P=\{x\in\mathbb{R}^n\mid Ax=b,x\geq0\}$ with rank$A=m<n$. For a ...
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1answer
34 views

How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
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Is it covex function?$J_{new}(u)=\int_{\Omega} \sum_{i=1}^{N} \lambda_if(x)u_i(x)dx$

I have a function such as $$J(u)=\int_{\Omega} \sum_{i=1}^{N} f(x)u_i(x)dx$$ where $f(x):\Omega \to R$, $0 \le u_i(x) \le 1,\sum_i u_i(x)=1$ Given that $J(u)$ is a convex function w.r.t $u$. Now I ...
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41 views

How does one evaluate the derivative of a matrix with a tensor $\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}$?

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
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72 views

How to show the convexity of this set

Is the set, $S=\{\bf x \in \mathbb{R}^n: \sum_{i=1}^{n} \frac{e^{x_i}}{1+e^{x_i}}=1 \}$, a convex set?
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13 views

Should the object function value be decreasing during the iteration procedure in ADMM

I want to solve the following convex optimization problem: $$\operatorname{argmin}\limits_X\|Ax-b\|_2^2+\lambda\sum_{i=1}^3 \|X_i\|_{*}$$ where $X$ is a three order tensor, $X_{(i)}$ is a matrix ...
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27 views

Compressive sensing for complex matrix

I'm fairly new to compressive sensing, and I have been looking for a MATLAB implementation of the problem $$ A x = b $$ where $A$ is non square, $x$ is kind of sparse and all the numbers involved are ...
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37 views

Linear independence of equality constraint gradients in constraint qualifications

I'm, trying to get an intuitive feel for the various constraint qualifications for KKT points. Most of them seem to rely on the linear independence of $\nabla g_i(x^*)$ where $g_i$ are the equality ...
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34 views

Optimization Example with some constraints !? [on hold]

We want to optimize the following function: $$ f(x,y)=x^2+3y^2+2xy+2 $$ with constraints $-2 \leq x< 2$, $-2 <y<2$, and $3y^2+x \leq 10$. Who can help me for the above example from ...
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11 views

the convergence of ADMM [on hold]

I solve a convex optimization problem by ADMM, I calculate the value of object function in each step ans plot it, why isn't the value increase decreasing? I don't know whether value of object ...
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41 views

Least Squares Nuclear Norm Optimization

I have the following least squares nuclear norm problem, $$ \min_{\bf X} \frac{1}{2}{\left\lVert {\bf b} - {{\bf W}}vec({\bf X}) \right\rVert}^2_2 + {\lambda_*}\Arrowvert {\bf X} \Arrowvert_* $$ ...
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Convergence results for block coordinate descent methods

I am trying to solve the problem minimize $f(x)$ subject to $x_1 \in C_1, x_2\in C_2, ... x_m\in C_m$ where $x_1, ..., x_m$ are block subvectors of $x$, and $C_i$ are each closed convex sets (not ...
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Accelerated Gradient Descent V.S Nonlinear Conjugate Gradient Descent

Let's consider smooth and convex minimization problem, i.e. $min f(x)$, where $f$ is not necessarily a quadratic function. If measured by iterations, Accelerated Gradient Descend (AGD) has ...
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1answer
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what are Smooth and Non Smooth Problem in optimization?

I am trying to understand the difference between the optimization problem types which are basically smooth and non smooth. I also found this question what does a smooth curve mean? I understand that ...
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22 views

non-linearity and non-convexity

I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has ...
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1answer
29 views

Matrix norm in the objective of an optimization problem

I am stuck with the following optimization problem from research. The optimization problem have the following objective function: $\|Q-H\|_\infty$. Here $Q$ is a PSD matrix and $H$ is a symmetric ...
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1answer
43 views

Can I solve a problem like a combination of PCA and compressed sensing?

$$ \underset{A,x}{\text{minimize}} \quad \lambda \left\| x \right\|_{1} + \left\| A \right\|_{*} $$ $$ D = A + Mx $$ Where $M \in \mathbb{R}^{n \times m}$, $x \in \mathbb{R}^{m \times z}$, $E=Mx \in ...
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Question about sub differentiability of convex function

I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less ...
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2answers
58 views

In linear programming, how to check whether a convex polyhedron is contained in another

Suppose we have two convex polyhedra $P_1=\{x\in \mathbb{R}^n \mid A_1 x \geq b_1 \}$ and $P_2=\{x\in \mathbb{R}^n \mid A_2 x \geq b_2 \}$ Is there a way to check whether $P_1 \subseteq P_2$? I was ...
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38 views

Minimizing a quadratic term

$\mathbf{x_1},\mathbf{x_2}$ are known and I need to solve the following objective wrt to one variable $\mathbf{y}$. The single constraint is $y(1,1)=1$. This is expressed as an inner product ...
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220 views

How to solve $\min \limits_{\mathbf{x}} \| \mathbf{Ax}-\mathbf{b} \|^2$?

Let $\mathbf{x}=[x_1,\ldots,x_K]$. I have the following optimization problem: \begin{array}{rl} \min \limits_{\mathbf{x}} & \| \mathbf{Ax}-\mathbf{b} \|^2 \\ \mbox{s.t.} & x_k\ge 0, \forall ...
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33 views

Minimum quadratic form value within a line?

If I have $x\in R^n , C\in R^{m\times n}, d\in R^m$, $m<n$, then $Cx=d$ is a linear manifold. And $P\in R^{n\times n}$, $P>0$, the quadratic form is $y=x^TPx$ Is there an analytical expression ...
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What's the difference between this proximal method and subgradient projection?

http://stanford.edu/~boyd/papers/pdf/prox_algs.pdf In the link above it is proposed that the nonsmooth separable resource allocation problem $$\min \sum f_i(x_i) \ \ \text{s.t.} \ \ \textbf{1}^Tx = ...
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1answer
18 views

Subgradient of the optimal value of a linear program with respect to its parameters

Consider the linear program $f(c)=\min\{c'x\mid x\in\mathbb{R}^n,Ax=b,x\geq0\}$. Are there any results on what is $\partial f/\partial b$?
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Is $K =\{ S: \exists \text{ positive diagonal} D, D^TSD \;\text{diagonally dominant}\}$ convex?

I am doing some convex cone optimization and wonder whether the following set $K_1$ is convex or not. Assume the following matrices are all in $\mathbb{R}^{n\times n}$ and symmetric. Let the set of ...
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26 views

Convex envelopes of bivariate functions

In order to convexify my nonlinear non-convex program I need convex envelopes for the function $(x/y)^2$, both x,y are positive. I am only aware of the convex envelopes of the type $xy$ from here ...
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how to find the edges emanating from a given vertex in a polyhedron

Suppose my polyhedron $P$ is defined as $P={ x\in \mathbb{R}^n \mid Ax=b, x\geq0 }$ I have $x_0$, which is a vertex of $P$. How to find the edges emanating from $x_0$? In other words, I want to find ...
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1answer
34 views

Is the expectation of log-concave function still log-concave?

I know the expectation preserves the concavity (or convexity), but I was wondering is it still true that the expectation of log-concave function still log-concave; to be more precise, Let ...
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1answer
31 views

The Legendre-Fenchel transform of $BV$ semi-norm

I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it... Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and ...
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6 views

Gomory's cut typical running time until the constraint is fractional

I was considering the following problem. Say we are given an linear programming problem $$ \max c^Tx $$ $$ Ax \le b $$ $$ x \ge 0$$ Where instead I consider $i^{th}$ the optimal solution $X_i$ of ...
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How do I prove that objective function is not convex

Here is my objective function. \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}}\hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...
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how do I enter this objective function in cvx [closed]

Following is my expression.$\mathbf{z}$ is a vector and $\mathbf{P_i}$ is a matrix. Also $i>0$ and is not fixed. I am not sure how to I enter this objective function in cvx? ...
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Convex signal reconstruction for convex generator function?

Let $f : \mathbb{R} \mapsto \mathbb{R}$ be a convex (not affine) function and suppose that $y = f(x)$. We want to reconstruct the input $x$ for a given $y$, and the standard approach would be to ...
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proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
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141 views

Rank reduction to satisfy Barvinok's upper bound & Rank of a set notation

After reading n times the four first sections of the 4th Chapter of J.Dattorro's book (Convex Optimization & Euclidean Distance Geometry). I am confused between yes or no, every extreme point of ...
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14 views

Rewriting a strictly convex quadratic equality constraint

I have the following constraint $$ x^TAx+b^Tx=c $$ where $A$ is a positive definite matrix. Is there any way to take advantage of the strict convexity of this expression to reformulate the constraint ...
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Find the nearest point of $y$ from a probability simplex. [duplicate]

I need to compute the nearest point of $x$ from a probability simplex. Formally, I want to ask if there is a close form for the solution to the following optimization problem: for $y\in \mathbb{R}^k$, ...
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1answer
25 views

How do I show that the set $S$ in $\mathbb R^3$ defined by linear inequalities is a $3$-simplex?

Consider the set $S$ in $\mathbb R^3$ defined by the inequalities: $x+y+z \ge 1$ $-x+y+z \le 1$ $x-y+z \le 1$ $x+y-z \le 1$ How can I show that $S$ is a $3$-simplex ? (Convex hull of $3 + 1$ ...
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1answer
23 views

Functional Lifting in Optimisation - Reference Request

I'm looking to learn about the use of (functional) lifting applied to a non-convex optimisation problem to give a (larger) convex problem. Unfortunately, I'm having a great deal of trouble finding ...
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46 views

Minimizing the max function

Suppose we have the single-variable function $$f(x) = \max_k \{f_k(x)\}$$ where each $f_k$ is convex and smooth (and known beforehand). We want to minimize it over some bounded interval. We can, in ...
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Find $U \subset D $ with $|U|$ minimal s.t $v = (1/2,1/2,1/2)$ belongs to the convex hull of $U$.

Consider the convex hull $\text {conv} \{e_1,e_2,e_3,(1,1/2,1/2),(1/2,1,1/2), (1/2,1/2,1)\}$ in $\mathbb R^3$ and the vector $v = (1/2,1/2,1/2)$. I want to compute $U \subset ...
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20 views

strengths and weaknesses of analytical method

I was wondering if anyone could suggest any books or paper that explain/discuss the advantages and drawbacks of analytical methods for optimization. Also, if we have a convex objective function ...