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

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Matrix equation from optimization problem

I am having a problem to find the solution to the following equation which has arisen as part of the solution of a (convex) optimization problem I am considering. $$\left(\frac{a}{n ...
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23 views

Strong convexity, non-smoothness, and directional derivative

I have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ that is (strongly) convex (say in $\mathbb{R}^n$), but not necessarily differentiable. It attains its minimum at $\mathbf{q}$. Given two ...
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7 views

concavity of functions of many variables

I have a function in many variables, the function is concave and non-increasing in each one of the variables, is the entire function concave?
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20 views

Newton-Raphson convergence for function $f(\gamma)$ with $\gamma \geq0$ constraint.

I am reading an (engineering) paper that in the part of their solution, They propose a 2-step iterative solution based on Newton-Raphson method for concave function $f$ as below : ...
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1answer
16 views

Gauss-Newton convergence for constant Hessian

If I use Gauss-Newton to solve a least square optimization problem and $\mathbf{J}^H\mathbf{J}$ is constant does it imply that I will reach the solution in one iteration?
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1answer
33 views

Non-vanishing of sub gradient near optimal solution

Consider the non-smooth optimization problem \begin{equation} \min_{x \in \mathbb{R}^n} f(x). \end{equation} To solve the above problem, I am suing subgradient descent \begin{equation} ...
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1answer
18 views

Partitioning in convex problem (variables in two subsets)

Consider the following problem from textbook Convex Optimization Algorithm p.10: \begin{equation} \begin{aligned} &{\text{min}} & & F(x)+G(y)\\ ...
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1answer
36 views

Why lower semicontinuity?

I'm reading a proof on the existence of a solution to a minimisation problem, but I'm stuck. I give a brief summary of the arguments up to the point at which I'm stuck(at the yellow box). ...
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1answer
21 views

Proving convergence of projected subgradient descent

Any idea how to sum the series $\sum_{t=1}^T \frac{1}{\sqrt{t}} (\|x_t -a\|^2 -\|x_{t+1}-a\|^2) $, where $a$ is any constant and you can assume $\|x_{T+1}-a\|=0$. This sum occured in proving ...
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1answer
30 views

Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
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0answers
19 views

Existence of lagrange multipliers with polyhedral constraints

I am working with a paper (Exact regularization of polyhedral norms, Schöpfer 2012) which states as a well-known fact that, if $f$ is a polyhedral norm, then for some $\mu^* > 0$ \begin{equation} ...
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31 views

optimization of a function with inequality constraint

I have a function to be maximized subject to constraints. I can write the primal Lagrange function as the following: (objective function WITH two constraints in the last two terms) $$L_P = ...
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1answer
65 views

Arc length function of a helix/spiral is convex?

Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc ...
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1answer
46 views

How to computer the proximal operator of a indicator function?

For $G_{1}(x) = I_{c}(x), c =\{x|Ax=b\}$, the Proximal operator is: $Prox_{\gamma G_{1}}(x) = Proj_{c}(x) = x + A^{T}(AA^{T})^{-1}(y-Ax)$ I hope to know, how to derivative this result. And, for ...
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1answer
35 views

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 ...
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1answer
50 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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0answers
52 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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17 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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1answer
42 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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3answers
73 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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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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1answer
53 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 ...
2
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0answers
21 views

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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25 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
40 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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0answers
19 views

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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1answer
42 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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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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1answer
63 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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0answers
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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2answers
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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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0answers
26 views

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
21 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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1answer
35 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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0answers
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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 ...
2
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1answer
58 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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9 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 ...
2
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0answers
35 views

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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2answers
38 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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1answer
31 views

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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0answers
17 views

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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1answer
39 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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0answers
32 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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17 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 ...
2
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1answer
224 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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13 views

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 $y\in \mathbb{R}^k$, ...
0
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1answer
30 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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49 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 ...